Optimal. Leaf size=80 \[ \frac{2 x \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{n/2} (c-a c x)^{\frac{n-2}{2}} \text{Hypergeometric2F1}\left (1,-\frac{n}{2},1-\frac{n}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{n} \]
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Rubi [A] time = 0.146342, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6176, 6181, 131} \[ \frac{2 x \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{n/2} (c-a c x)^{\frac{n-2}{2}} \, _2F_1\left (1,-\frac{n}{2};1-\frac{n}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 131
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac{n}{2}} \, dx &=\left (\left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} x^{1-\frac{n}{2}} (c-a c x)^{-1+\frac{n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{-1+\frac{n}{2}} x^{-1+\frac{n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (\frac{1}{x}\right )^{-1+\frac{n}{2}} (c-a c x)^{-1+\frac{n}{2}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{2 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{n/2} x (c-a c x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (1,-\frac{n}{2};1-\frac{n}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0256766, size = 78, normalized size = 0.98 \[ -\frac{2 \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} (c-a c x)^{n/2} \text{Hypergeometric2F1}\left (1,-\frac{n}{2},1-\frac{n}{2},\frac{2}{a x+1}\right )}{a c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.353, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+c \right ) ^{-1+{\frac{n}{2}}}\, dx \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{\left (-a c x + c\right )}^{\frac{1}{2} \, n - 1} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a c x + c\right )}^{\frac{1}{2} \, n - 1} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, x\right ) \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{\left (-a c x + c\right )}^{\frac{1}{2} \, n - 1} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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