Optimal. Leaf size=281 \[ \frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{7}{6} c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{1}{6} a c x \sqrt{a^2 c x^2+c}+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
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Rubi [A] time = 0.376018, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4950, 4946, 4958, 4954, 217, 206, 4930, 195} \[ \frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{7}{6} c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{1}{6} a c x \sqrt{a^2 c x^2+c}+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4946
Rule 4958
Rule 4954
Rule 217
Rule 206
Rule 4930
Rule 195
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x} \, dx &=c \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx+\left (a^2 c\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{1}{3} (a c) \int \sqrt{c+a^2 c x^2} \, dx+c^2 \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\left (a c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{1}{6} a c x \sqrt{c+a^2 c x^2}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{1}{6} \left (a c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{6} a c x \sqrt{c+a^2 c x^2}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{2 c^2 \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 )}{\sqrt{c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{1}{6} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{1}{6} a c x \sqrt{c+a^2 c x^2}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{2 c^2 \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 )}{\sqrt{c+a^2 c x^2}}-\frac{7}{6} c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.245586, size = 220, normalized size = 0.78 \[ \frac{c \sqrt{a^2 c x^2+c} \left (6 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-6 i \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-a x \sqrt{a^2 x^2+1}+2 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+8 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-\sinh ^{-1}(a x)+6 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-6 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+6 \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-6 \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )\right )}{6 \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.312, size = 174, normalized size = 0.6 \begin{align*}{\frac{c \left ( 2\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-ax+8\,\arctan \left ( ax \right ) \right ) }{6}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{c}{3}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 7\,i\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\,i{\it dilog} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\,i{\it dilog} \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \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{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )}{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{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}{\left (a x \right )}}{x}\, 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{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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