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

Optimal. Leaf size=67 \[ \frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )+\frac{1}{2} x^2 \tan ^{-1}(x)-\frac{x}{2}+\frac{1}{2} i \tan ^{-1}(x)^2+\frac{1}{2} \tan ^{-1}(x)+\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.155049, antiderivative size = 67, 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 \[ \frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )+\frac{1}{2} x^2 \tan ^{-1}(x)-\frac{x}{2}+\frac{1}{2} i \tan ^{-1}(x)^2+\frac{1}{2} \tan ^{-1}(x)+\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.80396, size = 49, normalized size = 0.73 \[ \frac{x^{2} \operatorname{atan}{\left (x \right )}}{2} - \frac{x}{2} + \log{\left (\frac{2 i}{- x + i} \right )} \operatorname{atan}{\left (x \right )} + \frac{i \operatorname{atan}^{2}{\left (x \right )}}{2} + \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{i \operatorname{Li}_{2}\left (\frac{- x - i}{- x + i}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*atan(x)/2 - x/2 + log(2*I/(-x + I))*atan(x) + I*atan(x)**2/2 + atan(x)/2 +
I*polylog(2, (-x - I)/(-x + I))/2

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Mathematica [A]  time = 0.0537018, size = 53, normalized size = 0.79 \[ \frac{1}{2} \left (-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(x)}\right )+\tan ^{-1}(x) \left (x^2+2 \log \left (1+e^{2 i \tan ^{-1}(x)}\right )+1\right )-x-i \tan ^{-1}(x)^2\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-x - I*ArcTan[x]^2 + ArcTan[x]*(1 + x^2 + 2*Log[1 + E^((2*I)*ArcTan[x])]) - I*P
olyLog[2, -E^((2*I)*ArcTan[x])])/2

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Maple [C]  time = 0.052, size = 128, normalized size = 1.9 \[{\frac{{x}^{2}\arctan \left ( x \right ) }{2}}-{\frac{\arctan \left ( x \right ) \ln \left ({x}^{2}+1 \right ) }{2}}-{\frac{x}{2}}+{\frac{\arctan \left ( x \right ) }{2}}-{\frac{i}{4}}\ln \left ({x}^{2}+1 \right ) \ln \left ( x-i \right ) +{\frac{i}{8}} \left ( \ln \left ( x-i \right ) \right ) ^{2}+{\frac{i}{4}}\ln \left ( x-i \right ) \ln \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +{\frac{i}{4}}\ln \left ({x}^{2}+1 \right ) \ln \left ( x+i \right ) -{\frac{i}{8}} \left ( \ln \left ( x+i \right ) \right ) ^{2}-{\frac{i}{4}}\ln \left ( x+i \right ) \ln \left ({\frac{i}{2}} \left ( x-i \right ) \right ) -{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( x-i \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*arctan(x)/(x^2+1),x)

[Out]

1/2*x^2*arctan(x)-1/2*arctan(x)*ln(x^2+1)-1/2*x+1/2*arctan(x)-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)*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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3*arctan(x)/(x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^3*arctan(x)/(x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \operatorname{atan}{\left (x \right )}}{x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3*atan(x)/(x**2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3*arctan(x)/(x^2 + 1), x)