Optimal. Leaf size=377 \[ \frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^2 x}+\frac{6 a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.584324, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4966, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4898, 4894} \[ \frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^2 x}+\frac{6 a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4944
Rule 4958
Rule 4956
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4898
Rule 4894
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}+\left (6 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=\frac{6 a}{c \sqrt{c+a^2 c x^2}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}+\frac{\left (3 a \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a}{c \sqrt{c+a^2 c x^2}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}+\frac{\left (3 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a}{c \sqrt{c+a^2 c x^2}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac{6 a \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 (6 a \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 (6 a \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{6 a}{c \sqrt{c+a^2 c x^2}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac{6 a \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{6 i a \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{6 i a \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 (6 i a \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 (6 i a \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{6 a}{c \sqrt{c+a^2 c x^2}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac{6 a \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{6 i a \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{6 i a \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 (6 a \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 (6 a \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{6 a}{c \sqrt{c+a^2 c x^2}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac{6 a \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{6 i a \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{6 i a \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{6 a \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{6 a \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 = 1.47256, size = 301, normalized size = 0.8 \[ \frac{a \left (12 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-12 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-12 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+12 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )+6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )-6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-\frac{2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3 \sin ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )}{a x}-2 a x \tan ^{-1}(a x)^3-6 \tan ^{-1}(a x)^2+12 a x \tan ^{-1}(a x)-\frac{1}{2} a x \tan ^{-1}(a x)^3 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+12\right )}{2 c \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.319, size = 356, normalized size = 0.9 \begin{align*} -{\frac{a \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}-6\,i \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 ) ^{3}-6\,\arctan \left ( ax \right ) -3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,i \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 ) ^{3}}{{c}^{2}x}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-3\,{\frac{a\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}{\sqrt{{a}^{2}{x}^{2}+1}{c}^{2}} \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) - \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{\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 ) -2\,{\it polylog} \left ( 3,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ) } \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 )^{3}}{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}^{3}{\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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