Optimal. Leaf size=295 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 n \sqrt {c-\frac {c}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (1-n) \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2^{\frac {1+n}{2}} \sqrt {c-\frac {c}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{a (1-n) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A]
time = 0.15, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6332, 6329,
130, 71, 98, 133} \begin {gather*} \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}} \, _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 {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}} \, _2F_1\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 98
Rule 130
Rule 133
Rule 6329
Rule 6332
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}} \text {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*}
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Mathematica [A]
time = 0.34, size = 146, normalized size = 0.49 \begin {gather*} \frac {a e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \left (a (1+n) \sqrt {1-\frac {1}{a^2 x^2}} x+2 e^{\coth ^{-1}(a x)} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-e^{2 \coth ^{-1}(a x)}\right )+2 e^{\coth ^{-1}(a x)} n \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};e^{2 \coth ^{-1}(a x)}\right )\right )}{(1+n) \left (-1+a^2 x^2\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \sqrt {c -\frac {c}{a^{2} x^{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-\frac {c}{a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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