Optimal. Leaf size=104 \[ \frac{\left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^3 (n+4)}-\frac{(n+3) \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^3 (n+2) (n+4)} \]
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Rubi [A] time = 0.149331, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6175, 6180, 79, 37} \[ \frac{\left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^3 (n+4)}-\frac{(n+3) \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^3 (n+2) (n+4)} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 79
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\frac{\int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^3 x^3} \, dx}{a^3 c^3}\\ &=\frac{\operatorname{Subst}\left (\int x \left (1-\frac{x}{a}\right )^{-3-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{a^3 c^3}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^3 (4+n)}-\frac{(3+n) \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^3 (4+n)}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^3 (4+n)}-\frac{(3+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^3 (2+n) (4+n)}\\ \end{align*}
Mathematica [A] time = 0.216626, size = 64, normalized size = 0.62 \[ \frac{(-a x+n+3) e^{n \coth ^{-1}(a x)} \left (\cosh \left (3 \coth ^{-1}(a x)\right )+\sinh \left (3 \coth ^{-1}(a x)\right )\right )}{a^2 c^3 (n+2) (n+4) x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 46, normalized size = 0.4 \begin{align*} -{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( ax-n-3 \right ) \left ( ax+1 \right ) }{ \left ( ax-1 \right ) ^{2}{c}^{3} \left ({n}^{2}+6\,n+8 \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62983, size = 259, normalized size = 2.49 \begin{align*} -\frac{{\left (a^{2} x^{2} +{\left (a n - 2 \, a\right )} x + n - 3\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{3} n^{2} - 6 \, a c^{3} n + 8 \, a c^{3} +{\left (a^{3} c^{3} n^{2} - 6 \, a^{3} c^{3} n + 8 \, a^{3} c^{3}\right )} x^{2} - 2 \,{\left (a^{2} c^{3} n^{2} - 6 \, a^{2} c^{3} n + 8 \, a^{2} c^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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