Optimal. Leaf size=166 \[ -\frac {2 (n+2) \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c^2 n}-\frac {(n+3) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{a c^2 (n+2)}+\frac {x \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6179, 129, 155, 12, 131} \[ -\frac {2 (n+2) \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c^2 n}-\frac {(n+3) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{a c^2 (n+2)}+\frac {x \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 131
Rule 155
Rule 6179
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c^2}+\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {2+n}{a}-\frac {x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^2 (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c^2}-\frac {a \operatorname {Subst}\left (\int \frac {(2+n)^2 \left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{a^2 x} \, dx,x,\frac {1}{x}\right )}{c^2 (2+n)}\\ &=-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^2 (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c^2}-\frac {(2+n) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x} \, dx,x,\frac {1}{x}\right )}{a c^2}\\ &=-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^2 (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c^2}-\frac {2 (2+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c^2 n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 113, normalized size = 0.68 \[ \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left (n (a x+1) (n (a x-1)+2 a x-3)-2 (n+2)^2 (a x-1) \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a x-1}{a x+1}\right )\right )}{a c^2 n (n+2) (a x-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{2}}\, dx \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^2} \,d x \]
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 \[ \frac {a^{2} \int \frac {x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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