Optimal. Leaf size=235 \[ -\frac{1}{2} c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-i c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+i c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )+\frac{1}{12} a^2 c^2 x^2+\frac{2}{3} c^2 \log \left (a^2 x^2+1\right )+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+a^2 c^2 x^2 \tan ^{-1}(a x)^2-\frac{3}{2} a c^2 x \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]
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Rubi [A] time = 0.514568, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4948, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260, 266, 43} \[ -\frac{1}{2} c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-i c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+i c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )+\frac{1}{12} a^2 c^2 x^2+\frac{2}{3} c^2 \log \left (a^2 x^2+1\right )+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+a^2 c^2 x^2 \tan ^{-1}(a x)^2-\frac{3}{2} a c^2 x \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4850
Rule 4988
Rule 4884
Rule 4994
Rule 6610
Rule 4852
Rule 4916
Rule 4846
Rule 260
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac{c^2 \tan ^{-1}(a x)^2}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^2+a^4 c^2 x^3 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^2}{x} \, dx+\left (2 a^2 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx\\ &=a^2 c^2 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\left (4 a c^2\right ) \int \frac{\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a^3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{2} \left (a^5 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=a^2 c^2 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\left (2 a c^2\right ) \int \tan ^{-1}(a x) \, dx+\left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{2} \left (a^3 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{2} \left (a^3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-2 a c^2 x \tan ^{-1}(a x)-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+c^2 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+\left (i a c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (i a c^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{2} \left (a c^2\right ) \int \tan ^{-1}(a x) \, dx-\frac{1}{2} \left (a c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^2 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{6} \left (a^4 c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} a c^2 x \tan ^{-1}(a x)-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+c^2 \log \left (1+a^2 x^2\right )-i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^2 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} \left (a^2 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{12} \left (a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{3}{2} a c^2 x \tan ^{-1}(a x)-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{3}{4} c^2 \log \left (1+a^2 x^2\right )-i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^2 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{12} \left (a^4 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{12} a^2 c^2 x^2-\frac{3}{2} a c^2 x \tan ^{-1}(a x)-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{2}{3} c^2 \log \left (1+a^2 x^2\right )-i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^2 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.315548, size = 218, normalized size = 0.93 \[ \frac{1}{24} c^2 \left (24 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+24 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-12 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )+2 a^2 x^2+16 \log \left (a^2 x^2+1\right )+6 a^4 x^4 \tan ^{-1}(a x)^2-4 a^3 x^3 \tan ^{-1}(a x)+24 a^2 x^2 \tan ^{-1}(a x)^2-36 a x \tan ^{-1}(a x)+16 i \tan ^{-1}(a x)^3+18 \tan ^{-1}(a x)^2+24 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-24 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-i \pi ^3+2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 3.003, size = 1173, normalized size = 5. \begin{align*} \text{result too large to display} \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*} 12 \, a^{6} c^{2} \int \frac{x^{6} \arctan \left (a x\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + a^{6} c^{2} \int \frac{x^{6} \log \left (a^{2} x^{2} + 1\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + a^{6} c^{2} \int \frac{x^{6} \log \left (a^{2} x^{2} + 1\right )}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} - 2 \, a^{5} c^{2} \int \frac{x^{5} \arctan \left (a x\right )}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + 36 \, a^{4} c^{2} \int \frac{x^{4} \arctan \left (a x\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + 3 \, a^{4} c^{2} \int \frac{x^{4} \log \left (a^{2} x^{2} + 1\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + 4 \, a^{4} c^{2} \int \frac{x^{4} \log \left (a^{2} x^{2} + 1\right )}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} - 8 \, a^{3} c^{2} \int \frac{x^{3} \arctan \left (a x\right )}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + 36 \, a^{2} c^{2} \int \frac{x^{2} \arctan \left (a x\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + \frac{1}{32} \, c^{2} \log \left (a^{2} x^{2} + 1\right )^{3} + \frac{1}{16} \,{\left (a^{4} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 12 \, c^{2} \int \frac{\arctan \left (a x\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} + c^{2} \int \frac{\log \left (a^{2} x^{2} + 1\right )^{2}}{16 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} - \frac{1}{64} \,{\left (a^{4} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} \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 (\frac{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}{x}, 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 \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x}\, dx + \int 2 a^{2} x \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{4} x^{3} \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 \frac{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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