Optimal. Leaf size=41 \[ -\frac{1}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2} \]
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Rubi [A] time = 0.0930027, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {4902, 4970, 4406, 12, 3299} \[ -\frac{1}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 4902
Rule 4970
Rule 4406
Rule 12
Rule 3299
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac{1}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-(2 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx\\ &=-\frac{1}{a 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 )}{a c^2}\\ &=-\frac{1}{a 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 )}{a c^2}\\ &=-\frac{1}{a 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 )}{a c^2}\\ &=-\frac{1}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.0641981, size = 34, normalized size = 0.83 \[ -\frac{\frac{1}{a^2 x^2 \tan ^{-1}(a x)+\tan ^{-1}(a x)}+\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 37, normalized size = 0.9 \begin{align*} -{\frac{2\,{\it Si} \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +\cos \left ( 2\,\arctan \left ( ax \right ) \right ) +1}{2\,a{c}^{2}\arctan \left ( ax \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right ) \int \frac{x}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )}\,{d x} + 1}{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.07139, size = 286, normalized size = 6.98 \begin{align*} \frac{{\left (-i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) +{\left (i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2}{2 \,{\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\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{1}{a^{4} x^{4} \operatorname{atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )} + \operatorname{atan}^{2}{\left (a x \right )}}\, 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{1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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