Optimal. Leaf size=39 \[ \frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.168539, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4971, 4970, 3299} \[ \frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4971
Rule 4970
Rule 3299
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.11586, size = 37, normalized size = 0.95 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 \left (c \left (a^2 x^2+1\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.296, size = 82, normalized size = 2.1 \begin{align*} -{\frac{{\it csgn} \left ( \arctan \left ( ax \right ) \right ) \pi }{2\,{c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\it Si} \left ( \arctan \left ( ax \right ) \right ) }{{c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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*} \int \frac{x}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} x}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )}, 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*} \int \frac{x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}{\left (a x \right )}}\, dx \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 )}^{\frac{3}{2}} \arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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