Optimal. Leaf size=45 \[ \frac{1}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0248004, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4894} \[ \frac{1}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4894
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac{1}{a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0451827, size = 38, normalized size = 0.84 \[ \frac{\sqrt{a^2 c x^2+c} \left (a x \tan ^{-1}(a x)+1\right )}{c^2 \left (a^3 x^2+a\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.231, size = 98, normalized size = 2.2 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) +i \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\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 = 1.02519, size = 55, normalized size = 1.22 \begin{align*} \frac{x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} c} + \frac{1}{\sqrt{a^{2} c x^{2} + c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31497, size = 88, normalized size = 1.96 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x \arctan \left (a x\right ) + 1\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \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*} \int \frac{\operatorname{atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38002, size = 55, normalized size = 1.22 \begin{align*} \frac{x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} c} + \frac{1}{\sqrt{a^{2} c x^{2} + c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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