Optimal. Leaf size=67 \[ \frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{a}+x \cot ^{-1}(a x)^2+\frac {i \cot ^{-1}(a x)^2}{a}-\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4847, 4921, 4855, 2402, 2315} \[ \frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^2+\frac {i \cot ^{-1}(a x)^2}{a}-\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 4847
Rule 4855
Rule 4921
Rubi steps
\begin {align*} \int \cot ^{-1}(a x)^2 \, dx &=x \cot ^{-1}(a x)^2+(2 a) \int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-2 \int \frac {\cot ^{-1}(a x)}{i-a x} \, dx\\ &=\frac {i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a}\\ &=\frac {i \cot ^{-1}(a x)^2}{a}+x \cot ^{-1}(a x)^2-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 56, normalized size = 0.84 \[ \frac {i \text {Li}_2\left (e^{2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x) \left ((a x+i) \cot ^{-1}(a x)-2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arccot}\left (a x\right )^{2}, x\right ) \]
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 \[ \int \operatorname {arccot}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 136, normalized size = 2.03 \[ x \mathrm {arccot}\left (a x \right )^{2}+\frac {i \mathrm {arccot}\left (a x \right )^{2}}{a}-\frac {2 \,\mathrm {arccot}\left (a x \right ) \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}-\frac {2 \,\mathrm {arccot}\left (a x \right ) \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}+\frac {2 i \polylog \left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}+\frac {2 i \polylog \left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a} \]
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 \[ \frac {1}{4} \, x \arctan \left (1, a x\right )^{2} + 12 \, a^{2} \int \frac {x^{2} \arctan \left (\frac {1}{a x}\right )^{2}}{16 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + a^{2} \int \frac {x^{2} \log \left (a^{2} x^{2} + 1\right )^{2}}{16 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 4 \, a^{2} \int \frac {x^{2} \log \left (a^{2} x^{2} + 1\right )}{16 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac {1}{16} \, x \log \left (a^{2} x^{2} + 1\right )^{2} + \frac {\arctan \left (a x\right )^{3}}{4 \, a} + \frac {3 \, \arctan \left (a x\right )^{2} \arctan \left (\frac {1}{a x}\right )}{4 \, a} + \frac {3 \, \arctan \left (a x\right ) \arctan \left (\frac {1}{a x}\right )^{2}}{4 \, a} + 8 \, a \int \frac {x \arctan \left (\frac {1}{a x}\right )}{16 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + \int \frac {\log \left (a^{2} x^{2} + 1\right )^{2}}{16 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 55, normalized size = 0.82 \[ \frac {-2\,\ln \left (1-{\mathrm {e}}^{\mathrm {acot}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,\mathrm {acot}\left (a\,x\right )+\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {acot}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}+{\mathrm {acot}\left (a\,x\right )}^2\,1{}\mathrm {i}}{a}+x\,{\mathrm {acot}\left (a\,x\right )}^2 \]
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 \[ \int \operatorname {acot}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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