Optimal. Leaf size=71 \[ \frac{\text{Unintegrable}\left (\frac{1}{\tan ^{-1}(a x)},x\right )}{a^3 c^2}-\frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{x}{a^3 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{x}{a^3 c^2 \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.332047, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{a^2}+\frac{\int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{a^2 c}\\ &=-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}+\frac{x}{a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^3}+\frac{\int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a}+\frac{\int \frac{1}{\tan ^{-1}(a x)} \, dx}{a^3 c^2}\\ &=-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}+\frac{x}{a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)} \, dx}{a^3 c^2}\\ &=-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}+\frac{x}{a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)} \, dx}{a^3 c^2}\\ &=-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}+\frac{x}{a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-2 \frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)} \, dx}{a^3 c^2}\\ &=-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}+\frac{x}{a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\text{Ci}\left (2 \tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)} \, dx}{a^3 c^2}\\ \end{align*}
Mathematica [A] time = 4.063, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.482, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x^{3} - \frac{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )}{\left (a^{2} \int \frac{x^{4}}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x} + 3 \, \int \frac{x^{2}}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )}{a c^{2}}}{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3}}{a^{4} x^{4} \operatorname{atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )} + \operatorname{atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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