Integrand size = 13, antiderivative size = 48 \[ \int \left (-e^{-x}+e^x\right )^n \, dx=-\frac {\left (-e^{-x}+e^x\right )^n \left (1-e^{2 x}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},1-\frac {n}{2},e^{2 x}\right )}{n} \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2320, 2057, 372, 371} \[ \int \left (-e^{-x}+e^x\right )^n \, dx=-\frac {\left (e^x-e^{-x}\right )^n \left (1-e^{2 x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {n}{2},1-\frac {n}{2},e^{2 x}\right )}{n} \]
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Rule 371
Rule 372
Rule 2057
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (-\frac {1}{x}+x\right )^n}{x} \, dx,x,e^x\right ) \\ & = \left (\left (e^x\right )^n \left (-e^{-x}+e^x\right )^n \left (-1+e^{2 x}\right )^{-n}\right ) \text {Subst}\left (\int x^{-1-n} \left (-1+x^2\right )^n \, dx,x,e^x\right ) \\ & = \left (\left (e^x\right )^n \left (-e^{-x}+e^x\right )^n \left (1-e^{2 x}\right )^{-n}\right ) \text {Subst}\left (\int x^{-1-n} \left (1-x^2\right )^n \, dx,x,e^x\right ) \\ & = -\frac {\left (-e^{-x}+e^x\right )^n \left (1-e^{2 x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {n}{2},1-\frac {n}{2},e^{2 x}\right )}{n} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \left (-e^{-x}+e^x\right )^n \, dx=\frac {\left (-e^{-x}+e^x\right )^n \left (-1+e^{2 x}\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},1-\frac {n}{2},e^{2 x}\right )}{n} \]
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\[\int \left (-{\mathrm e}^{-x}+{\mathrm e}^{x}\right )^{n}d x\]
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\[ \int \left (-e^{-x}+e^x\right )^n \, dx=\int { {\left (-e^{\left (-x\right )} + e^{x}\right )}^{n} \,d x } \]
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\[ \int \left (-e^{-x}+e^x\right )^n \, dx=\int \left (e^{x} - e^{- x}\right )^{n}\, dx \]
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\[ \int \left (-e^{-x}+e^x\right )^n \, dx=\int { {\left (-e^{\left (-x\right )} + e^{x}\right )}^{n} \,d x } \]
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\[ \int \left (-e^{-x}+e^x\right )^n \, dx=\int { {\left (-e^{\left (-x\right )} + e^{x}\right )}^{n} \,d x } \]
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Timed out. \[ \int \left (-e^{-x}+e^x\right )^n \, dx=\int {\left ({\mathrm {e}}^x-{\mathrm {e}}^{-x}\right )}^n \,d x \]
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