Optimal. Leaf size=47 \[ \frac{x \left (1-x^2\right )^3}{4 \left (x^2+1\right )^4}+\frac{3 x \left (1-x^2\right )}{8 \left (x^2+1\right )^2}+\frac{3}{8} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0163951, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {413, 21, 203} \[ \frac{x \left (1-x^2\right )^3}{4 \left (x^2+1\right )^4}+\frac{3 x \left (1-x^2\right )}{8 \left (x^2+1\right )^2}+\frac{3}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 413
Rule 21
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (-1+x^2\right )^4}{\left (1+x^2\right )^5} \, dx &=\frac{x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac{1}{8} \int \frac{\left (-1+x^2\right )^2 \left (6+6 x^2\right )}{\left (1+x^2\right )^4} \, dx\\ &=\frac{x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac{3}{4} \int \frac{\left (-1+x^2\right )^2}{\left (1+x^2\right )^3} \, dx\\ &=\frac{x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac{3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac{3}{16} \int \frac{2+2 x^2}{\left (1+x^2\right )^2} \, dx\\ &=\frac{x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac{3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac{3}{8} \int \frac{1}{1+x^2} \, dx\\ &=\frac{x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac{3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac{3}{8} \tan ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0124781, size = 41, normalized size = 0.87 \[ \frac{-5 x^7+3 x^5-3 x^3+3 \left (x^2+1\right )^4 \tan ^{-1}(x)+5 x}{8 \left (x^2+1\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 33, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ({x}^{2}+1 \right ) ^{4}} \left ( -{\frac{5\,{x}^{7}}{8}}+{\frac{3\,{x}^{5}}{8}}-{\frac{3\,{x}^{3}}{8}}+{\frac{5\,x}{8}} \right ) }+{\frac{3\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46019, size = 65, normalized size = 1.38 \begin{align*} -\frac{5 \, x^{7} - 3 \, x^{5} + 3 \, x^{3} - 5 \, x}{8 \,{\left (x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1\right )}} + \frac{3}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53216, size = 159, normalized size = 3.38 \begin{align*} -\frac{5 \, x^{7} - 3 \, x^{5} + 3 \, x^{3} - 3 \,{\left (x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1\right )} \arctan \left (x\right ) - 5 \, x}{8 \,{\left (x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.156707, size = 46, normalized size = 0.98 \begin{align*} - \frac{5 x^{7} - 3 x^{5} + 3 x^{3} - 5 x}{8 x^{8} + 32 x^{6} + 48 x^{4} + 32 x^{2} + 8} + \frac{3 \operatorname{atan}{\left (x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44375, size = 73, normalized size = 1.55 \begin{align*} \frac{3}{32} \, \pi \mathrm{sgn}\left (x\right ) - \frac{5 \,{\left (x - \frac{1}{x}\right )}^{3} + 12 \, x - \frac{12}{x}}{8 \,{\left ({\left (x - \frac{1}{x}\right )}^{2} + 4\right )}^{2}} + \frac{3}{16} \, \arctan \left (\frac{x^{2} - 1}{2 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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