An AMM-like integral $int_0^1frac{arctan x}xlnfrac{(1+x^2)^3}{(1+x)^2}dx$











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How can we evaluate $$I=int_0^1frac{arctan x}xlnfrac{(1+x^2)^3}{(1+x)^2}dx=0?$$




I tried substitution $x=frac{1-t}{1+t}$ and got
$$I=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} arctan frac{t-1}{t+1}}{t^2-1}dt\
=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} (arctan t-fracpi4)}{t^2-1}dt$$

I'm able to evaluate $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt$$
But I have no idea where to start with the rest one.










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  • Just wondering: Why do you want to calculate this (by hand)
    – klirk
    Nov 20 at 20:38










  • @klirk Just an interest.
    – Kemono Chen
    Nov 21 at 0:18










  • $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt=frac{pi^2}{48}$$
    – user178256
    Nov 21 at 11:00















up vote
7
down vote

favorite
7













How can we evaluate $$I=int_0^1frac{arctan x}xlnfrac{(1+x^2)^3}{(1+x)^2}dx=0?$$




I tried substitution $x=frac{1-t}{1+t}$ and got
$$I=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} arctan frac{t-1}{t+1}}{t^2-1}dt\
=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} (arctan t-fracpi4)}{t^2-1}dt$$

I'm able to evaluate $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt$$
But I have no idea where to start with the rest one.










share|cite|improve this question






















  • Just wondering: Why do you want to calculate this (by hand)
    – klirk
    Nov 20 at 20:38










  • @klirk Just an interest.
    – Kemono Chen
    Nov 21 at 0:18










  • $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt=frac{pi^2}{48}$$
    – user178256
    Nov 21 at 11:00













up vote
7
down vote

favorite
7









up vote
7
down vote

favorite
7






7






How can we evaluate $$I=int_0^1frac{arctan x}xlnfrac{(1+x^2)^3}{(1+x)^2}dx=0?$$




I tried substitution $x=frac{1-t}{1+t}$ and got
$$I=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} arctan frac{t-1}{t+1}}{t^2-1}dt\
=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} (arctan t-fracpi4)}{t^2-1}dt$$

I'm able to evaluate $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt$$
But I have no idea where to start with the rest one.










share|cite|improve this question














How can we evaluate $$I=int_0^1frac{arctan x}xlnfrac{(1+x^2)^3}{(1+x)^2}dx=0?$$




I tried substitution $x=frac{1-t}{1+t}$ and got
$$I=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} arctan frac{t-1}{t+1}}{t^2-1}dt\
=int_0^1frac{2 ln frac{2 (t^2+1)^3}{(t+1)^4} (arctan t-fracpi4)}{t^2-1}dt$$

I'm able to evaluate $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt$$
But I have no idea where to start with the rest one.







calculus integration definite-integrals






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asked Nov 20 at 9:20









Kemono Chen

2,050435




2,050435












  • Just wondering: Why do you want to calculate this (by hand)
    – klirk
    Nov 20 at 20:38










  • @klirk Just an interest.
    – Kemono Chen
    Nov 21 at 0:18










  • $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt=frac{pi^2}{48}$$
    – user178256
    Nov 21 at 11:00


















  • Just wondering: Why do you want to calculate this (by hand)
    – klirk
    Nov 20 at 20:38










  • @klirk Just an interest.
    – Kemono Chen
    Nov 21 at 0:18










  • $$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt=frac{pi^2}{48}$$
    – user178256
    Nov 21 at 11:00
















Just wondering: Why do you want to calculate this (by hand)
– klirk
Nov 20 at 20:38




Just wondering: Why do you want to calculate this (by hand)
– klirk
Nov 20 at 20:38












@klirk Just an interest.
– Kemono Chen
Nov 21 at 0:18




@klirk Just an interest.
– Kemono Chen
Nov 21 at 0:18












$$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt=frac{pi^2}{48}$$
– user178256
Nov 21 at 11:00




$$int_0^1frac{ln frac{2 (t^2+1)^3}{(t+1)^4}}{t^2-1}dt=frac{pi^2}{48}$$
– user178256
Nov 21 at 11:00










2 Answers
2






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oldest

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up vote
3
down vote



accepted










$text{A solution by Cornel Ioan Valean.}$ The problem is similar to the problem $textbf{AMM 12054}.$ Using the well-known result in $textbf{4.535.1}$ from $text{Table of Integrals, Series and Products}$ by I.S. Gradshteyn and I.M. Ryzhik:
begin{equation*}
int_0^1 frac{arctan(y x)}{1+y^2x}textrm{d}x=frac{1}{2y^2}arctan(y)log(1+y^2),
end{equation*}

We have:
begin{equation*}
frac{1}{2}int_0^1frac{arctan(y)log(1+y^2)}{y}=int_0^1left(int_0^1 frac{yarctan(y x)}{1+y^2x}textrm{d}xright)textrm{d}yoverset{yx=t}{=}int_0^1left(int_0^y frac{arctan(t)}{1+y t}textrm{d}tright)textrm{d}y\
end{equation*}

begin{equation*}
=int_0^1left(int_t^1 frac{arctan(t)}{1+y t}textrm{d}yright)textrm{d}t=int_0^1frac{displaystyle arctan(y)logleft(frac{1+y}{1+y^2}right)}{y} textrm{d}y,
end{equation*}

And the result is proved.






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  • 1




    (+1) Very neat.
    – nospoon
    Nov 21 at 12:24


















up vote
0
down vote













Through the dilogarithm/trilogarithm machinery it can be shown that



$$ int_{0}^{1}frac{log(1+i x)log(1+x)}{x},dx=\frac{pi K}{2}-frac{9ipi^3}{64}+3iKlog(2)-frac{3pi i}{16}log^2(2)+frac{5pi^2}{32}log(2)-frac{log^3(2)}{8}-frac{69}{16}zeta(3)+6,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log^2(1+i x)}{x},dx=\
-frac{pi K}{2}-frac{3ipi^3}{64}+iKlog(2)-frac{pi i}{16}log^2(2)+frac{5pi^2}{96}log(2)-frac{log^3(2)}{24}-frac{3}{16}zeta(3)+2,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log(1+ix)log(1-ix)}{x},dx= frac{pi K}{2}-frac{27}{32}zeta(3)$$
hence the claim follows by $arctan x=text{Im},log(1+ix)$ and $log(1+x^2)=log(1+ix)+log(1-ix)$.






share|cite|improve this answer





















  • Thank you for the great answer. :) But I prefer a solution without brute force.
    – Kemono Chen
    Nov 21 at 3:02











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2 Answers
2






active

oldest

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2 Answers
2






active

oldest

votes









active

oldest

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active

oldest

votes








up vote
3
down vote



accepted










$text{A solution by Cornel Ioan Valean.}$ The problem is similar to the problem $textbf{AMM 12054}.$ Using the well-known result in $textbf{4.535.1}$ from $text{Table of Integrals, Series and Products}$ by I.S. Gradshteyn and I.M. Ryzhik:
begin{equation*}
int_0^1 frac{arctan(y x)}{1+y^2x}textrm{d}x=frac{1}{2y^2}arctan(y)log(1+y^2),
end{equation*}

We have:
begin{equation*}
frac{1}{2}int_0^1frac{arctan(y)log(1+y^2)}{y}=int_0^1left(int_0^1 frac{yarctan(y x)}{1+y^2x}textrm{d}xright)textrm{d}yoverset{yx=t}{=}int_0^1left(int_0^y frac{arctan(t)}{1+y t}textrm{d}tright)textrm{d}y\
end{equation*}

begin{equation*}
=int_0^1left(int_t^1 frac{arctan(t)}{1+y t}textrm{d}yright)textrm{d}t=int_0^1frac{displaystyle arctan(y)logleft(frac{1+y}{1+y^2}right)}{y} textrm{d}y,
end{equation*}

And the result is proved.






share|cite|improve this answer



















  • 1




    (+1) Very neat.
    – nospoon
    Nov 21 at 12:24















up vote
3
down vote



accepted










$text{A solution by Cornel Ioan Valean.}$ The problem is similar to the problem $textbf{AMM 12054}.$ Using the well-known result in $textbf{4.535.1}$ from $text{Table of Integrals, Series and Products}$ by I.S. Gradshteyn and I.M. Ryzhik:
begin{equation*}
int_0^1 frac{arctan(y x)}{1+y^2x}textrm{d}x=frac{1}{2y^2}arctan(y)log(1+y^2),
end{equation*}

We have:
begin{equation*}
frac{1}{2}int_0^1frac{arctan(y)log(1+y^2)}{y}=int_0^1left(int_0^1 frac{yarctan(y x)}{1+y^2x}textrm{d}xright)textrm{d}yoverset{yx=t}{=}int_0^1left(int_0^y frac{arctan(t)}{1+y t}textrm{d}tright)textrm{d}y\
end{equation*}

begin{equation*}
=int_0^1left(int_t^1 frac{arctan(t)}{1+y t}textrm{d}yright)textrm{d}t=int_0^1frac{displaystyle arctan(y)logleft(frac{1+y}{1+y^2}right)}{y} textrm{d}y,
end{equation*}

And the result is proved.






share|cite|improve this answer



















  • 1




    (+1) Very neat.
    – nospoon
    Nov 21 at 12:24













up vote
3
down vote



accepted







up vote
3
down vote



accepted






$text{A solution by Cornel Ioan Valean.}$ The problem is similar to the problem $textbf{AMM 12054}.$ Using the well-known result in $textbf{4.535.1}$ from $text{Table of Integrals, Series and Products}$ by I.S. Gradshteyn and I.M. Ryzhik:
begin{equation*}
int_0^1 frac{arctan(y x)}{1+y^2x}textrm{d}x=frac{1}{2y^2}arctan(y)log(1+y^2),
end{equation*}

We have:
begin{equation*}
frac{1}{2}int_0^1frac{arctan(y)log(1+y^2)}{y}=int_0^1left(int_0^1 frac{yarctan(y x)}{1+y^2x}textrm{d}xright)textrm{d}yoverset{yx=t}{=}int_0^1left(int_0^y frac{arctan(t)}{1+y t}textrm{d}tright)textrm{d}y\
end{equation*}

begin{equation*}
=int_0^1left(int_t^1 frac{arctan(t)}{1+y t}textrm{d}yright)textrm{d}t=int_0^1frac{displaystyle arctan(y)logleft(frac{1+y}{1+y^2}right)}{y} textrm{d}y,
end{equation*}

And the result is proved.






share|cite|improve this answer














$text{A solution by Cornel Ioan Valean.}$ The problem is similar to the problem $textbf{AMM 12054}.$ Using the well-known result in $textbf{4.535.1}$ from $text{Table of Integrals, Series and Products}$ by I.S. Gradshteyn and I.M. Ryzhik:
begin{equation*}
int_0^1 frac{arctan(y x)}{1+y^2x}textrm{d}x=frac{1}{2y^2}arctan(y)log(1+y^2),
end{equation*}

We have:
begin{equation*}
frac{1}{2}int_0^1frac{arctan(y)log(1+y^2)}{y}=int_0^1left(int_0^1 frac{yarctan(y x)}{1+y^2x}textrm{d}xright)textrm{d}yoverset{yx=t}{=}int_0^1left(int_0^y frac{arctan(t)}{1+y t}textrm{d}tright)textrm{d}y\
end{equation*}

begin{equation*}
=int_0^1left(int_t^1 frac{arctan(t)}{1+y t}textrm{d}yright)textrm{d}t=int_0^1frac{displaystyle arctan(y)logleft(frac{1+y}{1+y^2}right)}{y} textrm{d}y,
end{equation*}

And the result is proved.







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edited Nov 21 at 12:19

























answered Nov 21 at 11:52









Zacky

3,2981337




3,2981337








  • 1




    (+1) Very neat.
    – nospoon
    Nov 21 at 12:24














  • 1




    (+1) Very neat.
    – nospoon
    Nov 21 at 12:24








1




1




(+1) Very neat.
– nospoon
Nov 21 at 12:24




(+1) Very neat.
– nospoon
Nov 21 at 12:24










up vote
0
down vote













Through the dilogarithm/trilogarithm machinery it can be shown that



$$ int_{0}^{1}frac{log(1+i x)log(1+x)}{x},dx=\frac{pi K}{2}-frac{9ipi^3}{64}+3iKlog(2)-frac{3pi i}{16}log^2(2)+frac{5pi^2}{32}log(2)-frac{log^3(2)}{8}-frac{69}{16}zeta(3)+6,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log^2(1+i x)}{x},dx=\
-frac{pi K}{2}-frac{3ipi^3}{64}+iKlog(2)-frac{pi i}{16}log^2(2)+frac{5pi^2}{96}log(2)-frac{log^3(2)}{24}-frac{3}{16}zeta(3)+2,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log(1+ix)log(1-ix)}{x},dx= frac{pi K}{2}-frac{27}{32}zeta(3)$$
hence the claim follows by $arctan x=text{Im},log(1+ix)$ and $log(1+x^2)=log(1+ix)+log(1-ix)$.






share|cite|improve this answer





















  • Thank you for the great answer. :) But I prefer a solution without brute force.
    – Kemono Chen
    Nov 21 at 3:02















up vote
0
down vote













Through the dilogarithm/trilogarithm machinery it can be shown that



$$ int_{0}^{1}frac{log(1+i x)log(1+x)}{x},dx=\frac{pi K}{2}-frac{9ipi^3}{64}+3iKlog(2)-frac{3pi i}{16}log^2(2)+frac{5pi^2}{32}log(2)-frac{log^3(2)}{8}-frac{69}{16}zeta(3)+6,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log^2(1+i x)}{x},dx=\
-frac{pi K}{2}-frac{3ipi^3}{64}+iKlog(2)-frac{pi i}{16}log^2(2)+frac{5pi^2}{96}log(2)-frac{log^3(2)}{24}-frac{3}{16}zeta(3)+2,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log(1+ix)log(1-ix)}{x},dx= frac{pi K}{2}-frac{27}{32}zeta(3)$$
hence the claim follows by $arctan x=text{Im},log(1+ix)$ and $log(1+x^2)=log(1+ix)+log(1-ix)$.






share|cite|improve this answer





















  • Thank you for the great answer. :) But I prefer a solution without brute force.
    – Kemono Chen
    Nov 21 at 3:02













up vote
0
down vote










up vote
0
down vote









Through the dilogarithm/trilogarithm machinery it can be shown that



$$ int_{0}^{1}frac{log(1+i x)log(1+x)}{x},dx=\frac{pi K}{2}-frac{9ipi^3}{64}+3iKlog(2)-frac{3pi i}{16}log^2(2)+frac{5pi^2}{32}log(2)-frac{log^3(2)}{8}-frac{69}{16}zeta(3)+6,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log^2(1+i x)}{x},dx=\
-frac{pi K}{2}-frac{3ipi^3}{64}+iKlog(2)-frac{pi i}{16}log^2(2)+frac{5pi^2}{96}log(2)-frac{log^3(2)}{24}-frac{3}{16}zeta(3)+2,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log(1+ix)log(1-ix)}{x},dx= frac{pi K}{2}-frac{27}{32}zeta(3)$$
hence the claim follows by $arctan x=text{Im},log(1+ix)$ and $log(1+x^2)=log(1+ix)+log(1-ix)$.






share|cite|improve this answer












Through the dilogarithm/trilogarithm machinery it can be shown that



$$ int_{0}^{1}frac{log(1+i x)log(1+x)}{x},dx=\frac{pi K}{2}-frac{9ipi^3}{64}+3iKlog(2)-frac{3pi i}{16}log^2(2)+frac{5pi^2}{32}log(2)-frac{log^3(2)}{8}-frac{69}{16}zeta(3)+6,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log^2(1+i x)}{x},dx=\
-frac{pi K}{2}-frac{3ipi^3}{64}+iKlog(2)-frac{pi i}{16}log^2(2)+frac{5pi^2}{96}log(2)-frac{log^3(2)}{24}-frac{3}{16}zeta(3)+2,text{Li}_3left(tfrac{1+i}{2}right) $$



$$ int_{0}^{1}frac{log(1+ix)log(1-ix)}{x},dx= frac{pi K}{2}-frac{27}{32}zeta(3)$$
hence the claim follows by $arctan x=text{Im},log(1+ix)$ and $log(1+x^2)=log(1+ix)+log(1-ix)$.







share|cite|improve this answer












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answered Nov 20 at 20:20









Jack D'Aurizio

285k33275654




285k33275654












  • Thank you for the great answer. :) But I prefer a solution without brute force.
    – Kemono Chen
    Nov 21 at 3:02


















  • Thank you for the great answer. :) But I prefer a solution without brute force.
    – Kemono Chen
    Nov 21 at 3:02
















Thank you for the great answer. :) But I prefer a solution without brute force.
– Kemono Chen
Nov 21 at 3:02




Thank you for the great answer. :) But I prefer a solution without brute force.
– Kemono Chen
Nov 21 at 3:02


















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