Optimal. Leaf size=49 \[ -\frac{x^2}{10 a^3}+\frac{\log \left (a^2 x^2+1\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x) \]
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Rubi [A] time = 0.0363023, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4853, 266, 43} \[ -\frac{x^2}{10 a^3}+\frac{\log \left (a^2 x^2+1\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4853
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^4 \cot ^{-1}(a x) \, dx &=\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{1}{5} a \int \frac{x^5}{1+a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{1}{10} a \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{1}{10} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{10 a^3}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{\log \left (1+a^2 x^2\right )}{10 a^5}\\ \end{align*}
Mathematica [A] time = 0.0123587, size = 49, normalized size = 1. \[ -\frac{x^2}{10 a^3}+\frac{\log \left (a^2 x^2+1\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 42, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{10\,{a}^{3}}}+{\frac{{x}^{4}}{20\,a}}+{\frac{{x}^{5}{\rm arccot} \left (ax\right )}{5}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{10\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97502, size = 62, normalized size = 1.27 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arccot}\left (a x\right ) + \frac{1}{20} \, a{\left (\frac{a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac{2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85585, size = 104, normalized size = 2.12 \begin{align*} \frac{4 \, a^{5} x^{5} \operatorname{arccot}\left (a x\right ) + a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} + 1\right )}{20 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19608, size = 46, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acot}{\left (a x \right )}}{5} + \frac{x^{4}}{20 a} - \frac{x^{2}}{10 a^{3}} + \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi x^{5}}{10} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08115, size = 68, normalized size = 1.39 \begin{align*} \frac{1}{5} \, x^{5} \arctan \left (\frac{1}{a x}\right ) + \frac{1}{20} \, a{\left (\frac{a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac{2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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