Optimal. Leaf size=46 \[ \frac{(1-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0923765, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {6186} \[ \frac{(1-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6186
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{e^{n \coth ^{-1}(a x)} (1-a n x)}{a^2 c \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.170015, size = 43, normalized size = 0.93 \[ \frac{(a n x-1) e^{n \coth ^{-1}(a x)}}{a^2 c \left (n^2-1\right ) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 49, normalized size = 1.1 \begin{align*} -{\frac{ \left ( nax-1 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{{a}^{2} \left ({n}^{2}-1 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \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 (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39743, size = 161, normalized size = 3.5 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a n x + 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} -{\left (a^{4} c^{2} n^{2} - a^{4} c^{2}\right )} x^{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{x e^{n \operatorname{acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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