Optimal. Leaf size=100 \[ -\frac{x}{4 c^2 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{2 a c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-\frac{\tan ^{-1}(a x)}{4 a c^2} \]
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Rubi [A] time = 0.0688284, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {4892, 4930, 199, 205} \[ -\frac{x}{4 c^2 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{2 a c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-\frac{\tan ^{-1}(a x)}{4 a c^2} \]
Antiderivative was successfully verified.
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Rule 4892
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-a \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac{\tan ^{-1}(a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-\frac{1}{2} \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=-\frac{x}{4 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-\frac{\int \frac{1}{c+a^2 c x^2} \, dx}{4 c}\\ &=-\frac{x}{4 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{4 a c^2}+\frac{\tan ^{-1}(a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}\\ \end{align*}
Mathematica [A] time = 0.0400243, size = 65, normalized size = 0.65 \[ \frac{2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3+\left (3-3 a^2 x^2\right ) \tan ^{-1}(a x)-3 a x+6 a x \tan ^{-1}(a x)^2}{12 c^2 \left (a^3 x^2+a\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 91, normalized size = 0.9 \begin{align*} -{\frac{x}{4\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{\arctan \left ( ax \right ) }{4\,a{c}^{2}}}+{\frac{\arctan \left ( ax \right ) }{2\,a{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{6\,a{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60479, size = 197, normalized size = 1.97 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{a^{2} c^{2} x^{2} + c^{2}} + \frac{\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2} + \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}}{12 \,{\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} - \frac{{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )}{2 \,{\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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Fricas [A] time = 2.12475, size = 162, normalized size = 1.62 \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 )}{12 \,{\left (a^{3} c^{2} x^{2} + a 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{\operatorname{atan}^{2}{\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{\arctan \left (a x\right )^{2}}{{\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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