Optimal. Leaf size=293 \[ \frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^2 x}+\frac{2 a^2 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.431192, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4966, 4944, 4958, 4954, 4898, 191} \[ \frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^2 x}+\frac{2 a^2 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4944
Rule 4958
Rule 4954
Rule 4898
Rule 191
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{c^2 x}+\left (2 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=\frac{2 a^2 x}{c \sqrt{c+a^2 c x^2}}-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{c^2 x}+\frac{\left (2 a \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a^2 x}{c \sqrt{c+a^2 c x^2}}-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{c^2 x}-\frac{4 a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{c+a^2 c x^2}}+\frac{2 i a \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{2 i a \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.04784, size = 226, normalized size = 0.77 \[ \frac{a \left (4 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-4 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )-\frac{2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2 \sin ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )}{a x}+4 a x-2 a x \tan ^{-1}(a x)^2-4 \tan ^{-1}(a x)-\frac{1}{2} a x \tan ^{-1}(a x)^2 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )}{2 c \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.306, size = 279, normalized size = 1. \begin{align*} -{\frac{a \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( ax+i \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) a}{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{{c}^{2}x}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{2\,ia}{{c}^{2}} \left ( i\arctan \left ( ax \right ) \ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{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 \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, 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 \frac{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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