Optimal. Leaf size=48 \[ -\frac{\left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^2 (n+2)} \]
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Rubi [A] time = 0.109206, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6175, 6180, 37} \[ -\frac{\left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac{\int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^2 x^2} \, dx}{a^2 c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{a^2 c^2}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^2 (2+n)}\\ \end{align*}
Mathematica [A] time = 0.167723, size = 33, normalized size = 0.69 \[ -\frac{(a x+1) e^{n \coth ^{-1}(a x)}}{a c^2 (n+2) (a x-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 33, normalized size = 0.7 \begin{align*} -{\frac{{{\rm e}^{n{\rm arccoth} \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 = 1.67622, size = 122, normalized size = 2.54 \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 [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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