Optimal. Leaf size=113 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right )}{2 a c^2}+\frac{a x}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2} \]
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Rubi [A] time = 0.259754, 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 = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c}\\ &=-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\left (2 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ &=-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ \end{align*}
Mathematica [A] time = 1.77278, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.275, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, 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{-a x +{\left (3 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) + \frac{2 \,{\left (a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )}{\left (3 \, a^{4} \int \frac{x^{4}}{a^{4} x^{7} \arctan \left (a x\right ) + 2 \, a^{2} x^{5} \arctan \left (a x\right ) + x^{3} \arctan \left (a x\right )}\,{d x} + 2 \, a^{2} \int \frac{x^{2}}{a^{4} x^{7} \arctan \left (a x\right ) + 2 \, a^{2} x^{5} \arctan \left (a x\right ) + x^{3} \arctan \left (a x\right )}\,{d x} + \int \frac{1}{{\left (a^{2} x^{2} + 1\right )}^{2} x^{3} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2}}{a^{2} c^{2}}}{2 \,{\left (a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right )^{2}} \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{1}{{\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \arctan \left (a x\right )^{3}}, 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{1}{a^{4} x^{5} \operatorname{atan}^{3}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname{atan}^{3}{\left (a x \right )} + x \operatorname{atan}^{3}{\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{1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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