Optimal. Leaf size=18 \[ \frac{e^{n \tanh ^{-1}(a x)}}{a c n} \]
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Rubi [A] time = 0.0343823, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6137} \[ \frac{e^{n \tanh ^{-1}(a x)}}{a c n} \]
Antiderivative was successfully verified.
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Rule 6137
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac{e^{n \tanh ^{-1}(a x)}}{a c n}\\ \end{align*}
Mathematica [A] time = 0.006812, size = 33, normalized size = 1.83 \[ \frac{(1-a x)^{-n/2} (a x+1)^{n/2}}{a c n} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.029, size = 18, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{can}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964448, size = 41, normalized size = 2.28 \begin{align*} \frac{e^{\left (\frac{1}{2} \, n \log \left (a x + 1\right ) - \frac{1}{2} \, n \log \left (a x - 1\right )\right )}}{a c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19292, size = 53, normalized size = 2.94 \begin{align*} \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c n} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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