Optimal. Leaf size=111 \[ \frac{1}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{1}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac{x \tan ^{-1}(a x)}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3} \]
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Rubi [A] time = 0.0748615, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4934, 4892, 261} \[ \frac{1}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{1}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac{x \tan ^{-1}(a x)}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3} \]
Antiderivative was successfully verified.
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Rule 4934
Rule 4892
Rule 261
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{1}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{\int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}\\ &=-\frac{1}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3}-\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=-\frac{1}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3}\\ \end{align*}
Mathematica [A] time = 0.0526421, size = 64, normalized size = 0.58 \[ \frac{a^2 x^2+2 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)+\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 101, normalized size = 0.9 \begin{align*}{\frac{\arctan \left ( ax \right ){x}^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{x\arctan \left ( ax \right ) }{8\,{a}^{2}{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,{a}^{3}{c}^{3}}}-{\frac{1}{16\,{a}^{3}{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{1}{16\,{a}^{3}{c}^{3} \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 = 1.70585, size = 174, normalized size = 1.57 \begin{align*} \frac{1}{8} \,{\left (\frac{a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac{\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right ) + \frac{{\left (a^{2} x^{2} -{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a}{16 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59638, size = 176, normalized size = 1.59 \begin{align*} \frac{a^{2} x^{2} +{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 2 \,{\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{16 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\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^{2} \operatorname{atan}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \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 )}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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