Optimal. Leaf size=111 \[ -\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{3 a^3}-\frac {\tan ^{-1}(a x)}{3 a^3}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac {x}{3 a^2}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {x^2 \cot ^{-1}(a x)}{3 a} \]
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Rubi [A] time = 0.14, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4853, 4917, 321, 203, 4921, 4855, 2402, 2315} \[ -\frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^3}+\frac {x}{3 a^2}-\frac {\tan ^{-1}(a x)}{3 a^3}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {x^2 \cot ^{-1}(a x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2315
Rule 2402
Rule 4853
Rule 4855
Rule 4917
Rule 4921
Rubi steps
\begin {align*} \int x^2 \cot ^{-1}(a x)^2 \, dx &=\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {1}{3} (2 a) \int \frac {x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2 \int x \cot ^{-1}(a x) \, dx}{3 a}-\frac {2 \int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}\\ &=\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {1}{3} \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {2 \int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{3 a^2}\\ &=\frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {\int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}\\ &=\frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\tan ^{-1}(a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3}\\ &=\frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\tan ^{-1}(a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 76, normalized size = 0.68 \[ \frac {\left (a^3 x^3-i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (a^2 x^2+2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )+1\right )-i \text {Li}_2\left (e^{2 i \cot ^{-1}(a x)}\right )+a x}{3 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arccot}\left (a x\right )^{2}, x\right ) \]
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 \[ \int x^{2} \operatorname {arccot}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 213, normalized size = 1.92 \[ \frac {x^{3} \mathrm {arccot}\left (a x \right )^{2}}{3}+\frac {x^{2} \mathrm {arccot}\left (a x \right )}{3 a}-\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3 a^{3}}+\frac {x}{3 a^{2}}-\frac {\arctan \left (a x \right )}{3 a^{3}}+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6 a^{3}}-\frac {i \ln \left (a x -i\right )^{2}}{12 a^{3}}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6 a^{3}}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6 a^{3}}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6 a^{3}}+\frac {i \ln \left (a x +i\right )^{2}}{12 a^{3}}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6 a^{3}}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{12} \, x^{3} \arctan \left (1, a x\right )^{2} - \frac {1}{48} \, x^{3} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {36 \, a^{2} x^{4} \arctan \left (1, a x\right )^{2} + 4 \, a^{2} x^{4} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a x^{3} \arctan \left (1, a x\right ) + 36 \, x^{2} \arctan \left (1, a x\right )^{2} + 3 \, {\left (a^{2} x^{4} + x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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