Integrand size = 22, antiderivative size = 289 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2+n)}+\frac {\left (6+4 n-n^2-n^3\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac {\left (6+4 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x}{c^2}+\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c^2} \]
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Time = 0.18 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6329, 105, 160, 12, 133} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {2 \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {n+2}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c^2}-\frac {\left (n^2+4 n+6\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-n/2}}{a c^2 n (n+2)}+\frac {\left (-n^3-n^2+4 n+6\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}}{a c^2 (2-n) n (n+2)}-\frac {(n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{a c^2 (n+2)}+\frac {x \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{c^2} \]
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Rule 12
Rule 105
Rule 133
Rule 160
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-2+\frac {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 {1}{2} (-2+n)} x}{c^2}+\frac {\text {Subst}\left (\int \frac {\left (-\frac {n}{a}-\frac {3 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-2+\frac {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 {1}{2} (-2+n)}}{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 {1}{2} (-2+n)} x}{c^2}-\frac {a \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-2+\frac {n}{2}} \left (\frac {n (2+n)}{a^2}+\frac {2 (3+n) x}{a^3}\right )}{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 {1}{2} (-2+n)}}{a c^2 (2+n)}-\frac {\left (6+4 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x}{c^2}+\frac {a^2 \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{-2+\frac {n}{2}} \left (-\frac {n^2 (2+n)}{a^3}-\frac {\left (6+4 n+n^2\right ) x}{a^4}\right )}{x} \, dx,x,\frac {1}{x}\right )}{c^2 n (2+n)} \\ & = -\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2+n)}+\frac {\left (6+4 n-n^2-n^3\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac {\left (6+4 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x}{c^2}-\frac {a^3 \text {Subst}\left (\int \frac {n^2 \left (4-n^2\right ) \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{-1+\frac {n}{2}}}{a^4 x} \, dx,x,\frac {1}{x}\right )}{c^2 n \left (4-n^2\right )} \\ & = -\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2+n)}+\frac {\left (6+4 n-n^2-n^3\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac {\left (6+4 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x}{c^2}-\frac {n \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{-1+\frac {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 {1}{2} (-2+n)}}{a c^2 (2+n)}+\frac {\left (6+4 n-n^2-n^3\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac {\left (6+4 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x}{c^2}+\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c^2} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.62 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (-6+n^2+6 a n x-a n^3 x+6 a^2 x^2-2 a^2 n^2 x^2-4 a^3 n x^3+a^3 n^3 x^3+e^{2 \coth ^{-1}(a x)} (-2+n) n^2 \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+n \left (-4+n^2\right ) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{a c^2 (-2+n) n (2+n) \left (-1+a^2 x^2\right )} \]
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\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{2}}d x\]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^2} \,d x \]
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