Optimal. Leaf size=61 \[ \frac{1}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2} \]
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Rubi [A] time = 0.0258874, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4892, 261} \[ \frac{1}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2} \]
Antiderivative was successfully verified.
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Rule 4892
Rule 261
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2}-\frac{1}{2} a \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac{1}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2}\\ \end{align*}
Mathematica [A] time = 0.0215124, size = 44, normalized size = 0.72 \[ \frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+1}{4 c^2 \left (a^3 x^2+a\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{4\,a{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{x\arctan \left ( ax \right ) }{2\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,a{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66636, size = 105, normalized size = 1.72 \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 ) - \frac{{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a}{4 \,{\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 = 1.65389, size = 109, normalized size = 1.79 \begin{align*} \frac{2 \, a x \arctan \left (a x\right ) +{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1}{4 \,{\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \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 )}{{\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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