Optimal. Leaf size=178 \[ -\frac{a^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c}+\frac{i a^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c}-\frac{a^2 \log \left (a^2 x^2+1\right )}{2 c}+\frac{a^2 \log (x)}{c}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}-\frac{a \tan ^{-1}(a x)}{c x} \]
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Rubi [A] time = 0.335515, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610} \[ -\frac{a^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c}+\frac{i a^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c}-\frac{a^2 \log \left (a^2 x^2+1\right )}{2 c}+\frac{a^2 \log (x)}{c}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}-\frac{a \tan ^{-1}(a x)}{c x} \]
Antiderivative was successfully verified.
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Rule 4918
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4884
Rule 4924
Rule 4868
Rule 4992
Rule 6610
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^3} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^2}{2 c x^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}+\frac{a \int \frac{\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c}-\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^2}{2 c x^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}+\frac{a \int \frac{\tan ^{-1}(a x)}{x^2} \, dx}{c}-\frac{a^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{c}+\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac{a \tan ^{-1}(a x)}{c x}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}+\frac{a^2 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx}{c}-\frac{\left (i a^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac{a \tan ^{-1}(a x)}{c x}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}-\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{a \tan ^{-1}(a x)}{c x}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}-\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 c}-\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{a \tan ^{-1}(a x)}{c x}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}+\frac{a^2 \log (x)}{c}-\frac{a^2 \log \left (1+a^2 x^2\right )}{2 c}-\frac{a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c}+\frac{i a^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}-\frac{a^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.305383, size = 142, normalized size = 0.8 \[ \frac{a^2 \left (-i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+\log \left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-\frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{2 a^2 x^2}-\frac{1}{3} i \tan ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{a x}-\tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+\frac{i \pi ^3}{24}\right )}{c} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.715, size = 5491, normalized size = 30.9 \begin{align*} \text{output 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*} \int \frac{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x^{3}}\,{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 (\frac{\arctan \left (a x\right )^{2}}{a^{2} c x^{5} + c x^{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*} \frac{\int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{a^{2} x^{5} + x^{3}}\, dx}{c} \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{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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