Optimal. Leaf size=41 \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{a^2 c^2}-\frac{x}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.210444, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4968, 4970, 3312, 3302, 4904} \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{a^2 c^2}-\frac{x}{a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4968
Rule 4970
Rule 3312
Rule 3302
Rule 4904
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx &=-\frac{x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a}-a \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx\\ &=-\frac{x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}-\frac{\operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}\\ &=-\frac{x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2}\\ &=-\frac{x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+2 \frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^2}\\ &=-\frac{x}{a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\text{Ci}\left (2 \tan ^{-1}(a x)\right )}{a^2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0696527, size = 36, normalized size = 0.88 \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )-\frac{a x}{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)}}{a^2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 38, normalized size = 0.9 \begin{align*}{\frac{2\,{\it Ci} \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{2\,{a}^{2}{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{x + \frac{{\left (a^{3} c^{2} x^{2} + a c^{2}\right )}{\left (a^{2} \int \frac{x^{2}}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x} - \int \frac{1}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )}{a c^{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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Fricas [C] time = 1.98861, size = 288, normalized size = 7.02 \begin{align*} \frac{{\left (a^{2} x^{2} + 1\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 (a^{2} x^{2} + 1\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 \, a x}{2 \,{\left (a^{4} c^{2} x^{2} + a^{2} 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{x}{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{x}{{\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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