Optimal. Leaf size=349 \[ \frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 x}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.341122, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4964, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191} \[ \frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 x}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4964
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4898
Rule 191
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac{\int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a^2 c}\\ &=-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{2 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{a^2 c \sqrt{c+a^2 c x^2}}\\ &=\frac{2 x}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt{c+a^2 c x^2}}\\ &=\frac{2 x}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{\left (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 )}{a^3 c \sqrt{c+a^2 c x^2}}+\frac{\left (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 )}{a^3 c \sqrt{c+a^2 c x^2}}\\ &=\frac{2 x}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt{c+a^2 c x^2}}+\frac{2 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 c \sqrt{c+a^2 c x^2}}-\frac{2 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 c \sqrt{c+a^2 c x^2}}-\frac{\left (2 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 c \sqrt{c+a^2 c x^2}}+\frac{\left (2 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 c \sqrt{c+a^2 c x^2}}\\ &=\frac{2 x}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt{c+a^2 c x^2}}+\frac{2 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 c \sqrt{c+a^2 c x^2}}-\frac{2 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 c \sqrt{c+a^2 c x^2}}-\frac{\left (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 )}{a^3 c \sqrt{c+a^2 c x^2}}+\frac{\left (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 )}{a^3 c \sqrt{c+a^2 c x^2}}\\ &=\frac{2 x}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{c+a^2 c x^2}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt{c+a^2 c x^2}}+\frac{2 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 c \sqrt{c+a^2 c x^2}}-\frac{2 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 c \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.347808, size = 228, normalized size = 0.65 \[ -\frac{\sqrt{a^2 x^2+1} \left (-2 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+2 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-\frac{2 a x}{\sqrt{a^2 x^2+1}}+\frac{a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{2 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\tan ^{-1}(a x)^2 \left (-\log \left (1-i e^{i \tan ^{-1}(a x)}\right )\right )+\tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )}{a^3 c \sqrt{c \left (a^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.873, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \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} x^{2} \arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{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{x^{2} \operatorname{atan}^{2}{\left (a x \right )}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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