Integrand size = 12, antiderivative size = 114 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}+\frac {2^{n/2} a^2 n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{2-n} \]
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Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6306, 81, 71} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\frac {a^2 2^{n/2} n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{2-n}+\frac {1}{2} a^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \]
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Rule 71
Rule 81
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}-\frac {1}{2} (a n) \text {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}+\frac {2^{n/2} a^2 n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{2-n} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {a^2 e^{n \coth ^{-1}(a x)} \left (-e^{2 \coth ^{-1}(a x)} n^2 \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+\frac {1}{a^2 x^2}+\frac {n}{a x}+n \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{2 (2+n)} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x^{3}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{3}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x^{3}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{3}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x^3} \,d x \]
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