Optimal. Leaf size=183 \[ \frac {2 n \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (1-n) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{\sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.17, antiderivative size = 183, normalized size of antiderivative = 1.00, 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 96
Rule 131
Rule 6194
Rule 6197
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*}
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Mathematica [A] time = 0.41, size = 112, normalized size = 0.61 \[ \frac {\left (a^2 x^2-1\right ) e^{n \coth ^{-1}(a x)} \left (a (n+1) x \sqrt {1-\frac {1}{a^2 x^2}}+2 n e^{\coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};e^{2 \coth ^{-1}(a x)}\right )\right )}{a^3 (n+1) x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a^{2} x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\sqrt {c -\frac {c}{a^{2} x^{2}}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a^{2} x^{2}}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{\sqrt {c-\frac {c}{a^2\,x^2}}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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