Optimal. Leaf size=242 \[ -\frac{4 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{10 a}-\frac{2 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{15 a}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{20 a^2}+\frac{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{8 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{1}{60} a c^2 x^3-\frac{11 c^2 x}{60 a}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a} \]
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Rubi [A] time = 0.188007, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4930, 4880, 4846, 4920, 4854, 2402, 2315, 8} \[ -\frac{4 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{10 a}-\frac{2 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{15 a}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{20 a^2}+\frac{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{8 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{1}{60} a c^2 x^3-\frac{11 c^2 x}{60 a}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4880
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 8
Rubi steps
\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{\int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx}{2 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{c \int \left (c+a^2 c x^2\right ) \, dx}{20 a}-\frac{(2 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx}{5 a}\\ &=-\frac{c^2 x}{20 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{\left (2 c^2\right ) \int 1 \, dx}{15 a}-\frac{\left (4 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx}{15 a}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{1}{15} \left (8 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{\left (8 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{15 a}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{8 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^2}+\frac{\left (8 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{15 a}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{8 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^2}-\frac{\left (8 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{15 a^2}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{8 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^2}-\frac{4 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^2}\\ \end{align*}
Mathematica [A] time = 0.743275, size = 131, normalized size = 0.54 \[ \frac{c^2 \left (16 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-a x \left (a^2 x^2+11\right )+10 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3-2 \left (3 a^5 x^5+10 a^3 x^3+15 a x-8 i\right ) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left (3 a^4 x^4+14 a^2 x^2-32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+11\right )\right )}{60 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.11, size = 368, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{6}}{6}}+{\frac{{a}^{2}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}}{2}}+{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}}{2}}-{\frac{{a}^{3}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}}{10}}-{\frac{a{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}}{3}}-{\frac{{c}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,a}}+{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{6\,{a}^{2}}}+{\frac{{a}^{2}{c}^{2}\arctan \left ( ax \right ){x}^{4}}{20}}+{\frac{7\,{c}^{2}\arctan \left ( ax \right ){x}^{2}}{30}}+{\frac{4\,{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{15\,{a}^{2}}}-{\frac{a{c}^{2}{x}^{3}}{60}}-{\frac{11\,{c}^{2}x}{60\,a}}+{\frac{11\,{c}^{2}\arctan \left ( ax \right ) }{60\,{a}^{2}}}-{\frac{{\frac{i}{15}}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{2}}}+{\frac{{\frac{i}{15}}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{2}}}+{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}}}-{\frac{{\frac{2\,i}{15}}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}}}+{\frac{{\frac{2\,i}{15}}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}}}+{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}-{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}}}-{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 ({\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \arctan \left (a x\right )^{3}, 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*} c^{2} \left (\int x \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{3} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{5} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \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{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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