Optimal. Leaf size=111 \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(1-n) \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.19688, antiderivative size = 111, 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 = {6192, 6195, 131} \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(1-n) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6195
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}} x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(1-n) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.176923, size = 81, normalized size = 0.73 \[ -\frac{2 \sqrt{c-a^2 c x^2} e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )}{\sqrt{1-\frac{1}{a^2 x^2}} \left (a^2 c n x+a^2 c x\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.306, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}}\, 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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{\sqrt{-a^{2} c x^{2} + c}}\,{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 (-\frac{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{acoth}{\left (a x \right )}}}{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \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}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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