3.648 \(\int \frac{\tan ^{-1}(x)^2}{x^5} \, dx\)

Optimal. Leaf size=61 \[ -\frac{\tan ^{-1}(x)^2}{4 x^4}-\frac{\tan ^{-1}(x)}{6 x^3}-\frac{1}{12 x^2}+\frac{1}{3} \log \left (x^2+1\right )-\frac{2 \log (x)}{3}+\frac{1}{4} \tan ^{-1}(x)^2+\frac{\tan ^{-1}(x)}{2 x} \]

[Out]

-1/(12*x^2) - ArcTan[x]/(6*x^3) + ArcTan[x]/(2*x) + ArcTan[x]^2/4 - ArcTan[x]^2/
(4*x^4) - (2*Log[x])/3 + Log[1 + x^2]/3

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Rubi [A]  time = 0.201209, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ -\frac{\tan ^{-1}(x)^2}{4 x^4}-\frac{\tan ^{-1}(x)}{6 x^3}-\frac{1}{12 x^2}+\frac{1}{3} \log \left (x^2+1\right )-\frac{2 \log (x)}{3}+\frac{1}{4} \tan ^{-1}(x)^2+\frac{\tan ^{-1}(x)}{2 x} \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[x]^2/x^5,x]

[Out]

-1/(12*x^2) - ArcTan[x]/(6*x^3) + ArcTan[x]/(2*x) + ArcTan[x]^2/4 - ArcTan[x]^2/
(4*x^4) - (2*Log[x])/3 + Log[1 + x^2]/3

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Rubi in Sympy [A]  time = 12.6269, size = 53, normalized size = 0.87 \[ - \frac{\log{\left (x^{2} \right )}}{3} + \frac{\log{\left (x^{2} + 1 \right )}}{3} + \frac{\operatorname{atan}^{2}{\left (x \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2 x} - \frac{1}{12 x^{2}} - \frac{\operatorname{atan}{\left (x \right )}}{6 x^{3}} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(atan(x)**2/x**5,x)

[Out]

-log(x**2)/3 + log(x**2 + 1)/3 + atan(x)**2/4 + atan(x)/(2*x) - 1/(12*x**2) - at
an(x)/(6*x**3) - atan(x)**2/(4*x**4)

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Mathematica [A]  time = 0.0133414, size = 56, normalized size = 0.92 \[ \frac{\left (x^4-1\right ) \tan ^{-1}(x)^2}{4 x^4}-\frac{1}{12 x^2}+\frac{1}{3} \log \left (x^2+1\right )+\frac{\left (3 x^2-1\right ) \tan ^{-1}(x)}{6 x^3}-\frac{2 \log (x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[x]^2/x^5,x]

[Out]

-1/(12*x^2) + ((-1 + 3*x^2)*ArcTan[x])/(6*x^3) + ((-1 + x^4)*ArcTan[x]^2)/(4*x^4
) - (2*Log[x])/3 + Log[1 + x^2]/3

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Maple [A]  time = 0.019, size = 48, normalized size = 0.8 \[ -{\frac{1}{12\,{x}^{2}}}-{\frac{\arctan \left ( x \right ) }{6\,{x}^{3}}}+{\frac{\arctan \left ( x \right ) }{2\,x}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}}-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4\,{x}^{4}}}-{\frac{2\,\ln \left ( x \right ) }{3}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x)^2/x^5,x)

[Out]

-1/12/x^2-1/6*arctan(x)/x^3+1/2*arctan(x)/x+1/4*arctan(x)^2-1/4*arctan(x)^2/x^4-
2/3*ln(x)+1/3*ln(x^2+1)

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Maxima [A]  time = 1.50342, size = 86, normalized size = 1.41 \[ \frac{1}{6} \,{\left (\frac{3 \, x^{2} - 1}{x^{3}} + 3 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) - \frac{3 \, x^{2} \arctan \left (x\right )^{2} - 4 \, x^{2} \log \left (x^{2} + 1\right ) + 8 \, x^{2} \log \left (x\right ) + 1}{12 \, x^{2}} - \frac{\arctan \left (x\right )^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x)^2/x^5,x, algorithm="maxima")

[Out]

1/6*((3*x^2 - 1)/x^3 + 3*arctan(x))*arctan(x) - 1/12*(3*x^2*arctan(x)^2 - 4*x^2*
log(x^2 + 1) + 8*x^2*log(x) + 1)/x^2 - 1/4*arctan(x)^2/x^4

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Fricas [A]  time = 0.232671, size = 72, normalized size = 1.18 \[ \frac{4 \, x^{4} \log \left (x^{2} + 1\right ) - 8 \, x^{4} \log \left (x\right ) + 3 \,{\left (x^{4} - 1\right )} \arctan \left (x\right )^{2} - x^{2} + 2 \,{\left (3 \, x^{3} - x\right )} \arctan \left (x\right )}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x)^2/x^5,x, algorithm="fricas")

[Out]

1/12*(4*x^4*log(x^2 + 1) - 8*x^4*log(x) + 3*(x^4 - 1)*arctan(x)^2 - x^2 + 2*(3*x
^3 - x)*arctan(x))/x^4

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Sympy [A]  time = 1.70025, size = 53, normalized size = 0.87 \[ - \frac{2 \log{\left (x \right )}}{3} + \frac{\log{\left (x^{2} + 1 \right )}}{3} + \frac{\operatorname{atan}^{2}{\left (x \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2 x} - \frac{1}{12 x^{2}} - \frac{\operatorname{atan}{\left (x \right )}}{6 x^{3}} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(atan(x)**2/x**5,x)

[Out]

-2*log(x)/3 + log(x**2 + 1)/3 + atan(x)**2/4 + atan(x)/(2*x) - 1/(12*x**2) - ata
n(x)/(6*x**3) - atan(x)**2/(4*x**4)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\arctan \left (x\right )^{2}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x)^2/x^5,x, algorithm="giac")

[Out]

integrate(arctan(x)^2/x^5, x)