Optimal. Leaf size=53 \[ \frac {\log \left (a^2 x^2+1\right )}{2 a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {x \cot ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4853, 4917, 4847, 260, 4885} \[ \frac {\log \left (a^2 x^2+1\right )}{2 a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {x \cot ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4847
Rule 4853
Rule 4885
Rule 4917
Rubi steps
\begin {align*} \int x \cot ^{-1}(a x)^2 \, dx &=\frac {1}{2} x^2 \cot ^{-1}(a x)^2+a \int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {\int \cot ^{-1}(a x) \, dx}{a}-\frac {\int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{a}\\ &=\frac {x \cot ^{-1}(a x)}{a}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\int \frac {x}{1+a^2 x^2} \, dx\\ &=\frac {x \cot ^{-1}(a x)}{a}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {\log \left (1+a^2 x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.79 \[ \frac {\log \left (a^2 x^2+1\right )+\left (a^2 x^2+1\right ) \cot ^{-1}(a x)^2+2 a x \cot ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 40, normalized size = 0.75 \[ \frac {2 \, a x \operatorname {arccot}\left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )^{2} + \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arccot}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 61, normalized size = 1.15 \[ \frac {x^{2} \mathrm {arccot}\left (a x \right )^{2}}{2}-\frac {\mathrm {arccot}\left (a x \right ) \arctan \left (a x \right )}{a^{2}}+\frac {x \,\mathrm {arccot}\left (a x \right )}{a}+\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 57, normalized size = 1.08 \[ \frac {1}{2} \, x^{2} \operatorname {arccot}\left (a x\right )^{2} + a {\left (\frac {x}{a^{2}} - \frac {\arctan \left (a x\right )}{a^{3}}\right )} \operatorname {arccot}\left (a x\right ) - \frac {\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 44, normalized size = 0.83 \[ \frac {x^2\,{\mathrm {acot}\left (a\,x\right )}^2}{2}+\frac {\frac {{\mathrm {acot}\left (a\,x\right )}^2}{2}+a\,x\,\mathrm {acot}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2+1\right )}{2}}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 54, normalized size = 1.02 \[ \begin {cases} \frac {x^{2} \operatorname {acot}^{2}{\left (a x \right )}}{2} + \frac {x \operatorname {acot}{\left (a x \right )}}{a} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a^{2}} + \frac {\operatorname {acot}^{2}{\left (a x \right )}}{2 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{2}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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