Integrand size = 23, antiderivative size = 75 \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1+\frac {1}{a x}\right )^{1+2 p} \operatorname {Hypergeometric2F1}\left (2,1+2 p,2 (1+p),1+\frac {1}{a x}\right )}{a (1+2 p)} \]
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Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6332, 6329, 67} \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{a x}+1\right )^{2 p+1} \left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (2,2 p+1,2 (p+1),1+\frac {1}{a x}\right )}{a (2 p+1)} \]
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Rule 67
Rule 6329
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p\right ) \int e^{2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p \, dx \\ & = -\left (\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{2 p}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1+\frac {1}{a x}\right )^{1+2 p} \operatorname {Hypergeometric2F1}\left (2,1+2 p,2 (1+p),1+\frac {1}{a x}\right )}{a (1+2 p)} \\ \end{align*}
\[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx \]
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\[\int {\mathrm e}^{2 p \,\operatorname {arccoth}\left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}d x\]
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\[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p} \,d x } \]
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\[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{2 p \operatorname {acoth}{\left (a x \right )}}\, dx \]
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\[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p} \,d x } \]
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\[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p} \,d x } \]
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Timed out. \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int {\mathrm {e}}^{2\,p\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]
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