3.674 \(\int \frac{x^3 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=79 \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )-\frac{x}{4 \left (x^2+1\right )}+\frac{\tan ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\frac{1}{4} \tan ^{-1}(x)-\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]

[Out]

-x/(4*(1 + x^2)) - ArcTan[x]/4 + ArcTan[x]/(2*(1 + x^2)) - (I/2)*ArcTan[x]^2 - A
rcTan[x]*Log[2/(1 + I*x)] - (I/2)*PolyLog[2, 1 - 2/(1 + I*x)]

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Rubi [A]  time = 0.177409, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )-\frac{x}{4 \left (x^2+1\right )}+\frac{\tan ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\frac{1}{4} \tan ^{-1}(x)-\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x^3*ArcTan[x])/(1 + x^2)^2,x]

[Out]

-x/(4*(1 + x^2)) - ArcTan[x]/4 + ArcTan[x]/(2*(1 + x^2)) - (I/2)*ArcTan[x]^2 - A
rcTan[x]*Log[2/(1 + I*x)] - (I/2)*PolyLog[2, 1 - 2/(1 + I*x)]

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Rubi in Sympy [A]  time = 10.5117, size = 56, normalized size = 0.71 \[ - \frac{x}{4 \left (x^{2} + 1\right )} - \log{\left (\frac{2 i}{- x + i} \right )} \operatorname{atan}{\left (x \right )} - \frac{i \operatorname{atan}^{2}{\left (x \right )}}{2} - \frac{\operatorname{atan}{\left (x \right )}}{4} - \frac{i \operatorname{Li}_{2}\left (\frac{- x - i}{- x + i}\right )}{2} + \frac{\operatorname{atan}{\left (x \right )}}{2 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*atan(x)/(x**2+1)**2,x)

[Out]

-x/(4*(x**2 + 1)) - log(2*I/(-x + I))*atan(x) - I*atan(x)**2/2 - atan(x)/4 - I*p
olylog(2, (-x - I)/(-x + I))/2 + atan(x)/(2*(x**2 + 1))

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Mathematica [A]  time = 0.0389717, size = 64, normalized size = 0.81 \[ \frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(x)}\right )+\frac{1}{2} i \tan ^{-1}(x)^2-\tan ^{-1}(x) \log \left (1+e^{2 i \tan ^{-1}(x)}\right )-\frac{1}{8} \sin \left (2 \tan ^{-1}(x)\right )+\frac{1}{4} \tan ^{-1}(x) \cos \left (2 \tan ^{-1}(x)\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*ArcTan[x])/(1 + x^2)^2,x]

[Out]

(I/2)*ArcTan[x]^2 + (ArcTan[x]*Cos[2*ArcTan[x]])/4 - ArcTan[x]*Log[1 + E^((2*I)*
ArcTan[x])] + (I/2)*PolyLog[2, -E^((2*I)*ArcTan[x])] - Sin[2*ArcTan[x]]/8

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Maple [C]  time = 0.045, size = 139, normalized size = 1.8 \[{\frac{\arctan \left ( x \right ) \ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2\,{x}^{2}+2}}+{\frac{i}{4}}\ln \left ({x}^{2}+1 \right ) \ln \left ( x-i \right ) -{\frac{i}{8}} \left ( \ln \left ( x-i \right ) \right ) ^{2}-{\frac{i}{4}}\ln \left ( x-i \right ) \ln \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) -{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) -{\frac{i}{4}}\ln \left ({x}^{2}+1 \right ) \ln \left ( x+i \right ) +{\frac{i}{8}} \left ( \ln \left ( x+i \right ) \right ) ^{2}+{\frac{i}{4}}\ln \left ( x+i \right ) \ln \left ({\frac{i}{2}} \left ( x-i \right ) \right ) +{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( x-i \right ) \right ) -{\frac{x}{4\,{x}^{2}+4}}-{\frac{\arctan \left ( x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*arctan(x)/(x^2+1)^2,x)

[Out]

1/2*arctan(x)*ln(x^2+1)+1/2*arctan(x)/(x^2+1)+1/4*I*ln(x^2+1)*ln(x-I)-1/8*I*ln(x
-I)^2-1/4*I*ln(x-I)*ln(-1/2*I*(x+I))-1/4*I*dilog(-1/2*I*(x+I))-1/4*I*ln(x^2+1)*l
n(x+I)+1/8*I*ln(x+I)^2+1/4*I*ln(x+I)*ln(1/2*I*(x-I))+1/4*I*dilog(1/2*I*(x-I))-1/
4*x/(x^2+1)-1/4*arctan(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \arctan \left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arctan(x)/(x^2 + 1)^2,x, algorithm="maxima")

[Out]

integrate(x^3*arctan(x)/(x^2 + 1)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{3} \arctan \left (x\right )}{x^{4} + 2 \, x^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arctan(x)/(x^2 + 1)^2,x, algorithm="fricas")

[Out]

integral(x^3*arctan(x)/(x^4 + 2*x^2 + 1), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*atan(x)/(x**2+1)**2,x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \arctan \left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arctan(x)/(x^2 + 1)^2,x, algorithm="giac")

[Out]

integrate(x^3*arctan(x)/(x^2 + 1)^2, x)