Optimal. Leaf size=127 \[ \frac {2^{\frac {n}{2}+1} \left (1-\frac {1}{a x}\right )^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{n}-\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{n} \]
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Rubi [A] time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6171, 105, 69, 131} \[ \frac {2^{\frac {n}{2}+1} \left (1-\frac {1}{a x}\right )^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{n}-\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 105
Rule 131
Rule 6171
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{a}-\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 )\\ &=-\frac {2 \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 )}{n}+\frac {2^{1+\frac {n}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 142, normalized size = 1.12 \[ \frac {e^{n \coth ^{-1}(a x)} \left (n e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right )+n e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \coth ^{-1}(a x)}\right )-(n+2) \left (\, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right )-\, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{n (n+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{x}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{x}\, dx \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x} \,d x \]
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 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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