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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]