Optimal. Leaf size=357 \[ -\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left (a^2 c x^2+c\right )^{5/2}}{30 a^3 c}+\frac{\left (a^2 c x^2+c\right )^{3/2}}{72 a^3}+\frac{c \sqrt{a^2 c x^2+c}}{16 a^3}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2} \]
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Rubi [A] time = 0.782542, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4950, 4946, 4952, 261, 4890, 4886, 266, 43} \[ -\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left (a^2 c x^2+c\right )^{5/2}}{30 a^3 c}+\frac{\left (a^2 c x^2+c\right )^{3/2}}{72 a^3}+\frac{c \sqrt{a^2 c x^2+c}}{16 a^3}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2} \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4946
Rule 4952
Rule 261
Rule 4890
Rule 4886
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx &=c \int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{4} c^2 \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{4} \left (a c^2\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{6} \left (a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{6} \left (a^3 c^2\right ) \int \frac{x^5}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac{7}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{8} c^2 \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}-\frac{c^2 \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{8 a}-\frac{1}{24} \left (a c^2\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{8} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{12} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{8 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac{7}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{c^2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{16 a^2}+\frac{c^2 \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{16 a}-\frac{1}{48} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{8} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac{1}{12} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{c+a^2 c x}}-\frac{2 \sqrt{c+a^2 c x}}{a^4 c}+\frac{\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{c \sqrt{c+a^2 c x^2}}{48 a^3}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{36 a^3}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a^3 c}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac{7}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{1}{48} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{16 a^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{c \sqrt{c+a^2 c x^2}}{16 a^3}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{72 a^3}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a^3 c}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac{7}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 5.94983, size = 576, normalized size = 1.61 \[ \frac{c \sqrt{a^2 c x^2+c} \left (-90 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+90 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\frac{3}{4} \left (a^2 x^2+1\right )^{5/2}+\frac{55}{8} \left (a^2 x^2+1\right )^3 \cos \left (3 \tan ^{-1}(a x)\right )-\frac{45}{8} \left (a^2 x^2+1\right )^3 \cos \left (5 \tan ^{-1}(a x)\right )+\frac{15}{16} \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x) \left (\frac{156 a x}{\sqrt{a^2 x^2+1}}+30 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-30 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-94 \sin \left (3 \tan ^{-1}(a x)\right )+6 \sin \left (5 \tan ^{-1}(a x)\right )+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )+45 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+18 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )\right )-\frac{15}{2} \left (a^2 x^2+1\right )^2 \left (-\frac{2}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x) \left (-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+\left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )\right )-6 \cos \left (3 \tan ^{-1}(a x)\right )\right )\right )}{1440 a^3 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.457, size = 221, normalized size = 0.6 \begin{align*}{\frac{c \left ( 120\,\arctan \left ( ax \right ){x}^{5}{a}^{5}-24\,{a}^{4}{x}^{4}+210\,\arctan \left ( ax \right ){x}^{3}{a}^{3}-38\,{a}^{2}{x}^{2}+45\,\arctan \left ( ax \right ) xa+31 \right ) }{720\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{c}{16\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\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^{4} + c x^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right ), 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 x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}{\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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