Optimal. Leaf size=128 \[ \frac{2 i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{3 a}+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{4 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{3 a}+\frac{c x}{3} \]
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Rubi [A] time = 0.0960557, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {4880, 4846, 4920, 4854, 2402, 2315, 8} \[ \frac{2 i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{3 a}+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac{c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{4 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{3 a}+\frac{c x}{3} \]
Antiderivative was successfully verified.
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Rule 4880
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 8
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{3} c \int 1 \, dx+\frac{1}{3} (2 c) \int \tan ^{-1}(a x)^2 \, dx\\ &=\frac{c x}{3}-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2-\frac{1}{3} (4 a c) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{c x}{3}-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{1}{3} (4 c) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx\\ &=\frac{c x}{3}-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{4 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a}-\frac{1}{3} (4 c) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{c x}{3}-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{4 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a}+\frac{(4 i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a}\\ &=\frac{c x}{3}-\frac{c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac{4 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a}+\frac{2 i c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0504754, size = 82, normalized size = 0.64 \[ \frac{c \left (-2 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+\left (a^3 x^3+3 a x-2 i\right ) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left (a^2 x^2-4 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+1\right )+a x\right )}{3 a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.086, size = 233, normalized size = 1.8 \begin{align*}{\frac{{a}^{2}c \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}}{3}}+cx \left ( \arctan \left ( ax \right ) \right ) ^{2}-{\frac{ac\arctan \left ( ax \right ){x}^{2}}{3}}-{\frac{2\,c\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,a}}+{\frac{cx}{3}}-{\frac{c\arctan \left ( ax \right ) }{3\,a}}-{\frac{{\frac{i}{3}}c\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{a}}+{\frac{{\frac{i}{6}}c \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{a}}+{\frac{{\frac{i}{3}}c\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{a}}+{\frac{{\frac{i}{3}}c{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{a}}+{\frac{{\frac{i}{3}}c\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{a}}-{\frac{{\frac{i}{6}}c \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{a}}-{\frac{{\frac{i}{3}}c\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{a}}-{\frac{{\frac{i}{3}}c{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{a}} \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*} 36 \, a^{4} c \int \frac{x^{4} \arctan \left (a x\right )^{2}}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{4} c \int \frac{x^{4} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 4 \, a^{4} c \int \frac{x^{4} \log \left (a^{2} x^{2} + 1\right )}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 8 \, a^{3} c \int \frac{x^{3} \arctan \left (a x\right )}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 72 \, a^{2} c \int \frac{x^{2} \arctan \left (a x\right )^{2}}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 6 \, a^{2} c \int \frac{x^{2} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{2} c \int \frac{x^{2} \log \left (a^{2} x^{2} + 1\right )}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac{1}{12} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \arctan \left (a x\right )^{2} + \frac{c \arctan \left (a x\right )^{3}}{4 \, a} - 24 \, a c \int \frac{x \arctan \left (a x\right )}{48 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac{1}{48} \,{\left (a^{2} c x^{3} + 3 \, c x\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + 3 \, c \int \frac{\log \left (a^{2} x^{2} + 1\right )^{2}}{48 \,{\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 ({\left (a^{2} c x^{2} + c\right )} \arctan \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*} c \left (\int a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int \operatorname{atan}^{2}{\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 )} \arctan \left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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