Optimal. Leaf size=60 \[ -\frac{1}{12 x^2}-\frac{1}{6} \log \left (x^2+1\right )-\frac{\left (x^2+1\right )^2 \tan ^{-1}(x)^2}{4 x^4}-\frac{\tan ^{-1}(x)}{6 x^3}+\frac{\log (x)}{3}-\frac{\tan ^{-1}(x)}{2 x} \]
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Rubi [A] time = 0.0779685, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {4944, 4950, 4852, 266, 44, 36, 29, 31} \[ -\frac{1}{12 x^2}-\frac{1}{6} \log \left (x^2+1\right )-\frac{\left (x^2+1\right )^2 \tan ^{-1}(x)^2}{4 x^4}-\frac{\tan ^{-1}(x)}{6 x^3}+\frac{\log (x)}{3}-\frac{\tan ^{-1}(x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 4944
Rule 4950
Rule 4852
Rule 266
Rule 44
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (1+x^2\right ) \tan ^{-1}(x)^2}{x^5} \, dx &=-\frac{\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac{1}{2} \int \frac{\left (1+x^2\right ) \tan ^{-1}(x)}{x^4} \, dx\\ &=-\frac{\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac{1}{2} \int \frac{\tan ^{-1}(x)}{x^4} \, dx+\frac{1}{2} \int \frac{\tan ^{-1}(x)}{x^2} \, dx\\ &=-\frac{\tan ^{-1}(x)}{6 x^3}-\frac{\tan ^{-1}(x)}{2 x}-\frac{\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac{1}{6} \int \frac{1}{x^3 \left (1+x^2\right )} \, dx+\frac{1}{2} \int \frac{1}{x \left (1+x^2\right )} \, dx\\ &=-\frac{\tan ^{-1}(x)}{6 x^3}-\frac{\tan ^{-1}(x)}{2 x}-\frac{\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,x^2\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac{\tan ^{-1}(x)}{6 x^3}-\frac{\tan ^{-1}(x)}{2 x}-\frac{\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac{1}{12} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,x^2\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )\\ &=-\frac{1}{12 x^2}-\frac{\tan ^{-1}(x)}{6 x^3}-\frac{\tan ^{-1}(x)}{2 x}-\frac{\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac{\log (x)}{3}-\frac{1}{6} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0318336, size = 56, normalized size = 0.93 \[ \frac{x^2 \left (4 x^2 \log (x)-2 x^2 \log \left (x^2+1\right )-1\right )-3 \left (x^2+1\right )^2 \tan ^{-1}(x)^2-2 \left (3 x^3+x\right ) \tan ^{-1}(x)}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 57, normalized size = 1. \begin{align*} -{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4\,{x}^{4}}}-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}}-{\frac{\arctan \left ( x \right ) }{6\,{x}^{3}}}-{\frac{\arctan \left ( x \right ) }{2\,x}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{6}}-{\frac{1}{12\,{x}^{2}}}+{\frac{\ln \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43972, size = 96, normalized size = 1.6 \begin{align*} -\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} - 2 \, x^{2} \log \left (x^{2} + 1\right ) + 4 \, x^{2} \log \left (x\right ) - 1}{12 \, x^{2}} - \frac{{\left (2 \, x^{2} + 1\right )} \arctan \left (x\right )^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42932, size = 153, normalized size = 2.55 \begin{align*} -\frac{2 \, x^{4} \log \left (x^{2} + 1\right ) - 4 \, x^{4} \log \left (x\right ) + 3 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right )^{2} + x^{2} + 2 \,{\left (3 \, x^{3} + x\right )} \arctan \left (x\right )}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.907907, size = 61, normalized size = 1.02 \begin{align*} \frac{\log{\left (x \right )}}{3} - \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{4} - \frac{\operatorname{atan}{\left (x \right )}}{2 x} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{2 x^{2}} - \frac{1}{12 x^{2}} - \frac{\operatorname{atan}{\left (x \right )}}{6 x^{3}} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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