Optimal. Leaf size=344 \[ -\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^3 c}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a^3 \sqrt{c}} \]
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Rubi [A] time = 0.333879, antiderivative size = 344, 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 = {4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589} \[ -\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^3 c}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a^3 \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 4952
Rule 4930
Rule 217
Rule 206
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^2}-\frac{\int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{\int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}-\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{a^2}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3 \sqrt{c}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3 \sqrt{c}}-\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (i \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 )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (i \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 )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3 \sqrt{c}}-\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3 \sqrt{c}}-\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.343681, size = 175, normalized size = 0.51 \[ \frac{\sqrt{a^2 c x^2+c} \left (\tan ^{-1}(a x) \left (a x \tan ^{-1}(a x)-2\right )+\frac{2 \left (-i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-\text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+\tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )}{\sqrt{a^2 x^2+1}}\right )}{2 a^3 c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.898, size = 271, normalized size = 0.8 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) xa-2 \right ) \arctan \left ( ax \right ) }{2\,c{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{1}{2\,c{a}^{3}} \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) - \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -4\,i\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,{\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -2\,{\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\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{x^{2} \arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c}}, 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{x^{2} \operatorname{atan}^{2}{\left (a x \right )}}{\sqrt{c \left (a^{2} x^{2} + 1\right )}}\, 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{x^{2} \arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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