Optimal. Leaf size=310 \[ \frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.50629, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4966, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4894} \[ \frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4958
Rule 4956
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4930
Rule 4894
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac{x \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 \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-(2 a) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{c \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{c \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}+\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{c \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}+\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{\left (2 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}+\frac{\left (2 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{c \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}+\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}+\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{c \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}+\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.308159, size = 204, normalized size = 0.66 \[ \frac{\sqrt{a^2 x^2+1} \left (2 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-2 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )-\frac{2}{\sqrt{a^2 x^2+1}}+\frac{\tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{2 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )-\tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )\right )}{c \sqrt{c \left (a^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.297, size = 307, normalized size = 1. \begin{align*}{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{1}{{c}^{2}} \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) - \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,{\it polylog} \left ( 3,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -2\,{\it polylog} \left ( 3,{\frac{1+iax}{\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^{5} + 2 \, a^{2} c^{2} x^{3} + c^{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*} \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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