Optimal. Leaf size=39 \[ -\frac {\log \left (a^2 x^2+1\right )}{6 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {x^2}{6 a} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \[ -\frac {\log \left (a^2 x^2+1\right )}{6 a^3}+\frac {x^2}{6 a}+\frac {1}{3} x^3 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4853
Rubi steps
\begin {align*} \int x^2 \cot ^{-1}(a x) \, dx &=\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {1}{3} a \int \frac {x^3}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {1}{6} a \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}{6 a}+\frac {1}{3} x^3 \cot ^{-1}(a x)-\frac {\log \left (1+a^2 x^2\right )}{6 a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.00 \[ -\frac {\log \left (a^2 x^2+1\right )}{6 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)+\frac {x^2}{6 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 37, normalized size = 0.95 \[ \frac {2 \, a^{3} x^{3} \operatorname {arccot}\left (a x\right ) + a^{2} x^{2} - \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 64, normalized size = 1.64 \[ \frac {1}{6} \, {\left (\frac {2 \, x^{3} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{2} {\left (\frac {1}{a^{2} x^{2}} - 1\right )}}{a^{2}} - \frac {\log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{4}} + \frac {\log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{4}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 0.87 \[ \frac {x^{2}}{6 a}+\frac {x^{3} \mathrm {arccot}\left (a x \right )}{3}-\frac {\ln \left (a^{2} x^{2}+1\right )}{6 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 0.92 \[ \frac {1}{3} \, x^{3} \operatorname {arccot}\left (a x\right ) + \frac {1}{6} \, a {\left (\frac {x^{2}}{a^{2}} - \frac {\log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 49, normalized size = 1.26 \[ \left \{\begin {array}{cl} \frac {\pi \,x^3}{6} & \text {\ if\ \ }a=0\\ \frac {\frac {x^2}{2}-\frac {\ln \left (a^2\,x^2+1\right )}{2\,a^2}}{3\,a}+\frac {x^3\,\mathrm {acot}\left (a\,x\right )}{3} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 37, normalized size = 0.95 \[ \begin {cases} \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {x^{2}}{6 a} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{6 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi x^{3}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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