Optimal. Leaf size=183 \[ \frac{2 n \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (1-n) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n+1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.165345, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6197, 6194, 96, 131} \[ \frac{2 n \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (1-n) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n+1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6194
Rule 96
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}}}\\ &=-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \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^2} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\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+n}{2}} x}{\sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\left (n \sqrt{1-\frac{1}{a^2 x^2}}\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 )}{a \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\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+n}{2}} x}{\sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 n \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)} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (1-n) \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.340543, size = 112, normalized size = 0.61 \[ \frac{\left (a^2 x^2-1\right ) e^{n \coth ^{-1}(a x)} \left (2 n e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )+a (n+1) x \sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^3 (n+1) x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.175, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{\frac{1}{\sqrt{c-{\frac{c}{{a}^{2}{x}^{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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{\sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{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{a^{2} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{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 (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\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{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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