Optimal. Leaf size=295 \[ \frac{2 n \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (1-n) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2^{\frac{n+1}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{a (1-n) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n+1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [C] time = 0.15462, antiderivative size = 111, normalized size of antiderivative = 0.38, 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 = {6197, 6194, 136} \[ -\frac{2^{\frac{3}{2}-\frac{n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n+3}{2}} F_1\left (\frac{n+3}{2};\frac{n-1}{2},2;\frac{n+5}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+3) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Rule 6197
Rule 6194
Rule 136
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int e^{n \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{\sqrt{c-\frac{c}{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{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{2^{\frac{3}{2}-\frac{n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} F_1\left (\frac{3+n}{2};\frac{1}{2} (-1+n),2;\frac{5+n}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (3+n) \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.463257, size = 146, normalized size = 0.49 \[ \frac{a x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a^2 x^2}} e^{n \coth ^{-1}(a x)} \left (2 e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \coth ^{-1}(a x)}\right )+2 n e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )+a (n+1) x \sqrt{1-\frac{1}{a^2 x^2}}\right )}{(n+1) \left (a^2 x^2-1\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}}\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 \sqrt{c - \frac{c}{a^{2} x^{2}}} \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 (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, 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 \sqrt{c - \frac{c}{a^{2} x^{2}}} \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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