Optimal. Leaf size=49 \[ \frac{x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0553146, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4930, 191} \[ \frac{x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 191
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=\frac{x}{a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0495469, size = 42, normalized size = 0.86 \[ \frac{\sqrt{a^2 c x^2+c} \left (a x-\tan ^{-1}(a x)\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.266, size = 100, normalized size = 2. \begin{align*} -{\frac{ \left ( \arctan \left ( ax \right ) +i \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74233, size = 38, normalized size = 0.78 \begin{align*} \frac{a x - \arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2285, size = 88, normalized size = 1.8 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x - \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3689, size = 61, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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