Optimal. Leaf size=44 \[ \frac{3 x}{32 \left (x^2+1\right )}+\frac{x}{16 \left (x^2+1\right )^2}-\frac{\tan ^{-1}(x)}{4 \left (x^2+1\right )^2}+\frac{3}{32} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0289532, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4930, 199, 203} \[ \frac{3 x}{32 \left (x^2+1\right )}+\frac{x}{16 \left (x^2+1\right )^2}-\frac{\tan ^{-1}(x)}{4 \left (x^2+1\right )^2}+\frac{3}{32} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4930
Rule 199
Rule 203
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(x)}{\left (1+x^2\right )^3} \, dx &=-\frac{\tan ^{-1}(x)}{4 \left (1+x^2\right )^2}+\frac{1}{4} \int \frac{1}{\left (1+x^2\right )^3} \, dx\\ &=\frac{x}{16 \left (1+x^2\right )^2}-\frac{\tan ^{-1}(x)}{4 \left (1+x^2\right )^2}+\frac{3}{16} \int \frac{1}{\left (1+x^2\right )^2} \, dx\\ &=\frac{x}{16 \left (1+x^2\right )^2}+\frac{3 x}{32 \left (1+x^2\right )}-\frac{\tan ^{-1}(x)}{4 \left (1+x^2\right )^2}+\frac{3}{32} \int \frac{1}{1+x^2} \, dx\\ &=\frac{x}{16 \left (1+x^2\right )^2}+\frac{3 x}{32 \left (1+x^2\right )}+\frac{3}{32} \tan ^{-1}(x)-\frac{\tan ^{-1}(x)}{4 \left (1+x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0204355, size = 36, normalized size = 0.82 \[ \frac{x \left (3 x^2+5\right )+\left (3 x^4+6 x^2-5\right ) \tan ^{-1}(x)}{32 \left (x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 37, normalized size = 0.8 \begin{align*}{\frac{x}{16\, \left ({x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x}{32\,{x}^{2}+32}}+{\frac{3\,\arctan \left ( x \right ) }{32}}-{\frac{\arctan \left ( x \right ) }{4\, \left ({x}^{2}+1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3996, size = 53, normalized size = 1.2 \begin{align*} \frac{3 \, x^{3} + 5 \, x}{32 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} - \frac{\arctan \left (x\right )}{4 \,{\left (x^{2} + 1\right )}^{2}} + \frac{3}{32} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43621, size = 95, normalized size = 2.16 \begin{align*} \frac{3 \, x^{3} +{\left (3 \, x^{4} + 6 \, x^{2} - 5\right )} \arctan \left (x\right ) + 5 \, x}{32 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.14173, size = 88, normalized size = 2. \begin{align*} \frac{3 x^{4} \operatorname{atan}{\left (x \right )}}{32 x^{4} + 64 x^{2} + 32} + \frac{3 x^{3}}{32 x^{4} + 64 x^{2} + 32} + \frac{6 x^{2} \operatorname{atan}{\left (x \right )}}{32 x^{4} + 64 x^{2} + 32} + \frac{5 x}{32 x^{4} + 64 x^{2} + 32} - \frac{5 \operatorname{atan}{\left (x \right )}}{32 x^{4} + 64 x^{2} + 32} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06281, size = 46, normalized size = 1.05 \begin{align*} \frac{3 \, x^{3} + 5 \, x}{32 \,{\left (x^{2} + 1\right )}^{2}} - \frac{\arctan \left (x\right )}{4 \,{\left (x^{2} + 1\right )}^{2}} + \frac{3}{32} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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