Optimal. Leaf size=111 \[ -\frac {2^{\frac {3}{2}-\frac {n}{2}} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} F_1\left (\frac {n+2}{2};\frac {n-1}{2},2;\frac {n+4}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2) \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.14, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {6182, 6179, 136} \[ -\frac {2^{\frac {3}{2}-\frac {n}{2}} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} F_1\left (\frac {n+2}{2};\frac {n-1}{2},2;\frac {n+4}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2) \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 136
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {2^{\frac {3}{2}-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \sqrt {c-\frac {c}{a x}} F_1\left (\frac {2+n}{2};\frac {1}{2} (-1+n),2;\frac {4+n}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [F] time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a c x - c}{a x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \sqrt {c -\frac {c}{a x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-\frac {c}{a\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- c \left (-1 + \frac {1}{a x}\right )} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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