Optimal. Leaf size=39 \[ \frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n+2}{2}}}{a c^2 (n+2)} \]
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Rubi [A] time = 0.0389181, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 37} \[ \frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n+2}{2}}}{a c^2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac{\int (1-a x)^{-2-\frac{n}{2}} (1+a x)^{n/2} \, dx}{c^2}\\ &=\frac{(1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^2 (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0191819, size = 39, normalized size = 1. \[ \frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}+1}}{a c^2 (n+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 33, normalized size = 0.9 \begin{align*} -{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( ax+1 \right ) }{ \left ( ax-1 \right ){c}^{2} \left ( 2+n \right ) a}} \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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a c x - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92778, size = 120, normalized size = 3.08 \begin{align*} \frac{{\left (a x + 1\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c^{2} n + 2 \, a c^{2} -{\left (a^{2} c^{2} n + 2 \, a^{2} c^{2}\right )} x} \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 = 1.11933, size = 43, normalized size = 1.1 \begin{align*} -\frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n + 1}}{a c^{2}{\left (n + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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