Optimal. Leaf size=135 \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{5 a^5}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}-\frac{3 x}{10 a^4}+\frac{3 \tan ^{-1}(a x)}{10 a^5}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}-\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{x^4 \cot ^{-1}(a x)}{10 a} \]
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Rubi [A] time = 0.210867, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {4853, 4917, 302, 203, 321, 4921, 4855, 2402, 2315} \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{5 a^5}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}-\frac{3 x}{10 a^4}+\frac{3 \tan ^{-1}(a x)}{10 a^5}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}-\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{x^4 \cot ^{-1}(a x)}{10 a} \]
Antiderivative was successfully verified.
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Rule 4853
Rule 4917
Rule 302
Rule 203
Rule 321
Rule 4921
Rule 4855
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x^4 \cot ^{-1}(a x)^2 \, dx &=\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{1}{5} (2 a) \int \frac{x^5 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{2 \int x^3 \cot ^{-1}(a x) \, dx}{5 a}-\frac{2 \int \frac{x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}\\ &=\frac{x^4 \cot ^{-1}(a x)}{10 a}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{1}{10} \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{2 \int x \cot ^{-1}(a x) \, dx}{5 a^3}+\frac{2 \int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^3}\\ &=-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}+\frac{x^4 \cot ^{-1}(a x)}{10 a}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{1}{10} \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{2 \int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{5 a^4}-\frac{\int \frac{x^2}{1+a^2 x^2} \, dx}{5 a^2}\\ &=-\frac{3 x}{10 a^4}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}+\frac{x^4 \cot ^{-1}(a x)}{10 a}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2-\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{5 a^5}+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{10 a^4}+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{5 a^4}-\frac{2 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^4}\\ &=-\frac{3 x}{10 a^4}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}+\frac{x^4 \cot ^{-1}(a x)}{10 a}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{3 \tan ^{-1}(a x)}{10 a^5}-\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{5 a^5}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^5}\\ &=-\frac{3 x}{10 a^4}+\frac{x^3}{30 a^2}-\frac{x^2 \cot ^{-1}(a x)}{5 a^3}+\frac{x^4 \cot ^{-1}(a x)}{10 a}+\frac{i \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^2+\frac{3 \tan ^{-1}(a x)}{10 a^5}-\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{5 a^5}+\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^5}\\ \end{align*}
Mathematica [A] time = 0.501046, size = 95, normalized size = 0.7 \[ \frac{6 i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+a x \left (a^2 x^2-9\right )+6 \left (a^5 x^5+i\right ) \cot ^{-1}(a x)^2+3 \cot ^{-1}(a x) \left (a^4 x^4-2 a^2 x^2-4 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )-3\right )}{30 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.121, size = 233, normalized size = 1.7 \begin{align*}{\frac{{x}^{5} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{5}}+{\frac{{x}^{4}{\rm arccot} \left (ax\right )}{10\,a}}-{\frac{{x}^{2}{\rm arccot} \left (ax\right )}{5\,{a}^{3}}}+{\frac{{\rm arccot} \left (ax\right )\ln \left ({a}^{2}{x}^{2}+1 \right ) }{5\,{a}^{5}}}+{\frac{{x}^{3}}{30\,{a}^{2}}}-{\frac{3\,x}{10\,{a}^{4}}}+{\frac{3\,\arctan \left ( ax \right ) }{10\,{a}^{5}}}+{\frac{{\frac{i}{20}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{5}}}+{\frac{{\frac{i}{10}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{5}}}-{\frac{{\frac{i}{10}}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{5}}}+{\frac{{\frac{i}{10}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{5}}}-{\frac{{\frac{i}{20}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{5}}}-{\frac{{\frac{i}{10}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{5}}}+{\frac{{\frac{i}{10}}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{5}}}-{\frac{{\frac{i}{10}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{20} \, x^{5} \arctan \left (1, a x\right )^{2} - \frac{1}{80} \, x^{5} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{60 \, a^{2} x^{6} \arctan \left (1, a x\right )^{2} + 4 \, a^{2} x^{6} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a x^{5} \arctan \left (1, a x\right ) + 60 \, x^{4} \arctan \left (1, a x\right )^{2} + 5 \,{\left (a^{2} x^{6} + x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{80 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{arccot}\left (a x\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{acot}^{2}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arccot}\left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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