Optimal. Leaf size=80 \[ \frac{x^2}{12 a^2}-\frac{\log \left (a^2 x^2+1\right )}{3 a^4}-\frac{x \cot ^{-1}(a x)}{2 a^3}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{x^3 \cot ^{-1}(a x)}{6 a} \]
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Rubi [A] time = 0.145659, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4853, 4917, 266, 43, 4847, 260, 4885} \[ \frac{x^2}{12 a^2}-\frac{\log \left (a^2 x^2+1\right )}{3 a^4}-\frac{x \cot ^{-1}(a x)}{2 a^3}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{x^3 \cot ^{-1}(a x)}{6 a} \]
Antiderivative was successfully verified.
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Rule 4853
Rule 4917
Rule 266
Rule 43
Rule 4847
Rule 260
Rule 4885
Rubi steps
\begin{align*} \int x^3 \cot ^{-1}(a x)^2 \, dx &=\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{1}{2} a \int \frac{x^4 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{\int x^2 \cot ^{-1}(a x) \, dx}{2 a}-\frac{\int \frac{x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a}\\ &=\frac{x^3 \cot ^{-1}(a x)}{6 a}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{1}{6} \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{\int \cot ^{-1}(a x) \, dx}{2 a^3}+\frac{\int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^3}\\ &=-\frac{x \cot ^{-1}(a x)}{2 a^3}+\frac{x^3 \cot ^{-1}(a x)}{6 a}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2+\frac{1}{12} \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{\int \frac{x}{1+a^2 x^2} \, dx}{2 a^2}\\ &=-\frac{x \cot ^{-1}(a x)}{2 a^3}+\frac{x^3 \cot ^{-1}(a x)}{6 a}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2-\frac{\log \left (1+a^2 x^2\right )}{4 a^4}+\frac{1}{12} \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{12 a^2}-\frac{x \cot ^{-1}(a x)}{2 a^3}+\frac{x^3 \cot ^{-1}(a x)}{6 a}-\frac{\cot ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^2-\frac{\log \left (1+a^2 x^2\right )}{3 a^4}\\ \end{align*}
Mathematica [A] time = 0.0203256, size = 61, normalized size = 0.76 \[ \frac{a^2 x^2-4 \log \left (a^2 x^2+1\right )+2 a x \left (a^2 x^2-3\right ) \cot ^{-1}(a x)+3 \left (a^4 x^4-1\right ) \cot ^{-1}(a x)^2}{12 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 82, normalized size = 1. \begin{align*}{\frac{{x}^{4} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{4}}+{\frac{{x}^{3}{\rm arccot} \left (ax\right )}{6\,a}}-{\frac{x{\rm arccot} \left (ax\right )}{2\,{a}^{3}}}+{\frac{{\rm arccot} \left (ax\right )\arctan \left ( ax \right ) }{2\,{a}^{4}}}+{\frac{{x}^{2}}{12\,{a}^{2}}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{a}^{4}}}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54347, size = 104, normalized size = 1.3 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccot}\left (a x\right )^{2} + \frac{1}{6} \, a{\left (\frac{a^{2} x^{3} - 3 \, x}{a^{4}} + \frac{3 \, \arctan \left (a x\right )}{a^{5}}\right )} \operatorname{arccot}\left (a x\right ) + \frac{a^{2} x^{2} + 3 \, \arctan \left (a x\right )^{2} - 4 \, \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91668, size = 144, normalized size = 1.8 \begin{align*} \frac{a^{2} x^{2} + 3 \,{\left (a^{4} x^{4} - 1\right )} \operatorname{arccot}\left (a x\right )^{2} + 2 \,{\left (a^{3} x^{3} - 3 \, a x\right )} \operatorname{arccot}\left (a x\right ) - 4 \, \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3269, size = 78, normalized size = 0.98 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acot}^{2}{\left (a x \right )}}{4} + \frac{x^{3} \operatorname{acot}{\left (a x \right )}}{6 a} + \frac{x^{2}}{12 a^{2}} - \frac{x \operatorname{acot}{\left (a x \right )}}{2 a^{3}} - \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{3 a^{4}} - \frac{\operatorname{acot}^{2}{\left (a x \right )}}{4 a^{4}} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x^{4}}{16} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13697, size = 146, normalized size = 1.82 \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (\frac{1}{a x}\right )^{2} + \frac{4 \, a^{3} i x^{3} \log \left (\frac{a x - i}{a x + i}\right ) + 4 \, a^{2} x^{2} - 12 \, a i x \log \left (\frac{a x - i}{a x + i}\right ) + 3 \, \log \left (\frac{a x - i}{a x + i}\right )^{2} - 16 \, \log \left (a^{2} x^{2} + 1\right )}{48 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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