Optimal. Leaf size=48 \[ -\frac {\left (1-e^{2 x}\right ) \left (e^x-e^{-x}\right )^n \, _2F_1\left (1,\frac {n+2}{2};1-\frac {n}{2};e^{2 x}\right )}{n} \]
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Rubi [A] time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2282, 2032, 365, 364} \[ -\frac {\left (e^x-e^{-x}\right )^n \left (1-e^{2 x}\right )^{-n} \text {Hypergeometric2F1}\left (-n,-\frac {n}{2},1-\frac {n}{2},e^{2 x}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2032
Rule 2282
Rubi steps
\begin {align*} \int \left (-e^{-x}+e^x\right )^n \, dx &=\operatorname {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 ) \operatorname {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 ) \operatorname {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} \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};e^{2 x}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 45, normalized size = 0.94 \[ \frac {\left (e^{2 x}-1\right ) \left (e^x-e^{-x}\right )^n \, _2F_1\left (1,\frac {n}{2}+1;1-\frac {n}{2};e^{2 x}\right )}{n} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (-e^{-x}+e^x\right )^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-e^{\left (-x\right )} + e^{x}\right )}^{n}, x\right ) \]
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 \[ \int {\left (-e^{\left (-x\right )} + e^{x}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \left (-{\mathrm e}^{-x}+{\mathrm e}^{x}\right )^{n}\, dx\]
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 \[ \int {\left (-e^{\left (-x\right )} + e^{x}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left ({\mathrm {e}}^x-{\mathrm {e}}^{-x}\right )}^n \,d x \]
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 \[ \int \left (e^{x} - e^{- x}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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