Optimal. Leaf size=133 \[ -\frac{3 x}{8 a c^2 \left (a^2 x^2+1\right )}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{3 \tan ^{-1}(a x)}{8 a^2 c^2} \]
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Rubi [A] time = 0.12161, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4930, 4892, 199, 205} \[ -\frac{3 x}{8 a c^2 \left (a^2 x^2+1\right )}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{3 \tan ^{-1}(a x)}{8 a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4892
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a}\\ &=\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3}{2} \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=-\frac{3 x}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \int \frac{1}{c+a^2 c x^2} \, dx}{8 a c}\\ &=-\frac{3 x}{8 a c^2 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)}{8 a^2 c^2}+\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0416157, size = 68, normalized size = 0.51 \[ \frac{2 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^3+\left (3-3 a^2 x^2\right ) \tan ^{-1}(a x)-3 a x+6 a x \tan ^{-1}(a x)^2}{8 a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 122, normalized size = 0.9 \begin{align*} -{\frac{3\,x}{8\,a{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{3\,\arctan \left ( ax \right ) }{8\,{a}^{2}{c}^{2}}}+{\frac{3\,\arctan \left ( ax \right ) }{4\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\,x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,a{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4\,{a}^{2}{c}^{2}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{2\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63228, size = 235, normalized size = 1.77 \begin{align*} \frac{3 \,{\left (\frac{x}{a^{2} c x^{2} + c} + \frac{\arctan \left (a x\right )}{a c}\right )} \arctan \left (a x\right )^{2}}{4 \, a c} + \frac{\frac{{\left (2 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{5} c x^{2} + a^{3} c} - \frac{6 \,{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )}{a^{4} c x^{2} + a^{2} c}}{8 \, a c} - \frac{\arctan \left (a x\right )^{3}}{2 \,{\left (a^{2} c x^{2} + c\right )} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99894, size = 163, normalized size = 1.23 \begin{align*} \frac{6 \, a x \arctan \left (a x\right )^{2} + 2 \,{\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \,{\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{8 \,{\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\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*} \frac{\int \frac{x \operatorname{atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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 \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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