Optimal. Leaf size=438 \[ \frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt{a^2 c x^2+c}}+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{6 a}+\frac{1}{12} c x \sqrt{a^2 c x^2+c}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{6 a} \]
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Rubi [A] time = 0.311359, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195} \[ \frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt{a^2 c x^2+c}}+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{6 a}+\frac{1}{12} c x \sqrt{a^2 c x^2+c}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{6 a} \]
Antiderivative was successfully verified.
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Rule 4880
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 217
Rule 206
Rule 195
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{6} c \int \sqrt{c+a^2 c x^2} \, dx+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{12} c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{4} \left (3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{12} c^2 \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}{4} \left (3 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 (3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt{c+a^2 c x^2}}+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a}-\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt{c+a^2 c x^2}}+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{\left (3 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{\left (3 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt{c+a^2 c x^2}}+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{12} c x \sqrt{c+a^2 c x^2}-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt{c+a^2 c x^2}}+\frac{5 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{3 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{3 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.935191, size = 439, normalized size = 1. \[ \frac{c \sqrt{a^2 c x^2+c} \left (72 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-72 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-72 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+72 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+2 a^3 x^3 \sqrt{a^2 x^2+1}+2 a x \sqrt{a^2 x^2+1}+21 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+2 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+69 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-94 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+80 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+2 a^4 x^4 \sin \left (3 \tan ^{-1}(a x)\right )-3 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+4 a^2 x^2 \sin \left (3 \tan ^{-1}(a x)\right )-6 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+6 a^4 x^4 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )+12 a^2 x^2 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-72 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+6 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )\right )}{96 a \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.317, size = 304, normalized size = 0.7 \begin{align*}{\frac{c \left ( 6\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}-4\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+15\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa+2\,ax-22\,\arctan \left ( ax \right ) \right ) }{24\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{i}{24}}c}{a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +18\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -18\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +18\,i{\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -18\,i{\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -40\,\arctan \left ({\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 ({\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{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 \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{2}{\left (a x \right )}\, 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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