Optimal. Leaf size=40 \[ \frac{x^2}{6}-\frac{2}{3} \log \left (x^2+1\right )+\frac{1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2 \]
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Rubi [A] time = 0.0965447, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {4917, 4853, 266, 43, 4847, 260, 4885} \[ \frac{x^2}{6}-\frac{2}{3} \log \left (x^2+1\right )+\frac{1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 4917
Rule 4853
Rule 266
Rule 43
Rule 4847
Rule 260
Rule 4885
Rubi steps
\begin{align*} \int \frac{x^4 \cot ^{-1}(x)}{1+x^2} \, dx &=\int x^2 \cot ^{-1}(x) \, dx-\int \frac{x^2 \cot ^{-1}(x)}{1+x^2} \, dx\\ &=\frac{1}{3} x^3 \cot ^{-1}(x)+\frac{1}{3} \int \frac{x^3}{1+x^2} \, dx-\int \cot ^{-1}(x) \, dx+\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx\\ &=-x \cot ^{-1}(x)+\frac{1}{3} x^3 \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2+\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,x^2\right )-\int \frac{x}{1+x^2} \, dx\\ &=-x \cot ^{-1}(x)+\frac{1}{3} x^3 \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2-\frac{1}{2} \log \left (1+x^2\right )+\frac{1}{6} \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{6}-x \cot ^{-1}(x)+\frac{1}{3} x^3 \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2-\frac{2}{3} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0274571, size = 32, normalized size = 0.8 \[ \frac{1}{6} \left (x^2-4 \log \left (x^2+1\right )+2 \left (x^2-3\right ) x \cot ^{-1}(x)-3 \cot ^{-1}(x)^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 38, normalized size = 1. \begin{align*}{\frac{{x}^{3}{\rm arccot} \left (x\right )}{3}}-x{\rm arccot} \left (x\right )+{\rm arccot} \left (x\right )\arctan \left ( x \right ) +{\frac{{x}^{2}}{6}}-{\frac{2\,\ln \left ({x}^{2}+1 \right ) }{3}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47917, size = 47, normalized size = 1.18 \begin{align*} \frac{1}{6} \, x^{2} + \frac{1}{3} \,{\left (x^{3} - 3 \, x + 3 \, \arctan \left (x\right )\right )} \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \arctan \left (x\right )^{2} - \frac{2}{3} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99925, size = 100, normalized size = 2.5 \begin{align*} \frac{1}{6} \, x^{2} + \frac{1}{3} \,{\left (x^{3} - 3 \, x\right )} \operatorname{arccot}\left (x\right ) - \frac{1}{2} \, \operatorname{arccot}\left (x\right )^{2} - \frac{2}{3} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.710818, size = 34, normalized size = 0.85 \begin{align*} \frac{x^{3} \operatorname{acot}{\left (x \right )}}{3} + \frac{x^{2}}{6} - x \operatorname{acot}{\left (x \right )} - \frac{2 \log{\left (x^{2} + 1 \right )}}{3} - \frac{\operatorname{acot}^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13736, size = 90, normalized size = 2.25 \begin{align*} \frac{1}{6} \, i x^{3} \log \left (-\frac{i - x}{i + x}\right ) - \frac{1}{2} \, i x \log \left (-\frac{i - x}{i + x}\right ) + \frac{1}{6} \, x^{2} + \frac{1}{8} \, \log \left (-\frac{i - x}{i + x}\right )^{2} - \frac{2}{3} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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