3.500 \(\int (-e^{-x}+e^x)^n \, dx\)

Optimal. Leaf size=48 \[ -\frac{\left (1-e^{2 x}\right ) \left (e^x-e^{-x}\right )^n \, _2F_1\left (1,\frac{n+2}{2};1-\frac{n}{2};e^{2 x}\right )}{n} \]

[Out]

-(((-E^(-x) + E^x)^n*(1 - E^(2*x))*Hypergeometric2F1[1, (2 + n)/2, 1 - n/2, E^(2*x)])/n)

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Rubi [A]  time = 0.0470604, 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.

[In]

Int[(-E^(-x) + E^x)^n,x]

[Out]

-(((-E^(-x) + E^x)^n*Hypergeometric2F1[-n, -n/2, 1 - n/2, E^(2*x)])/((1 - E^(2*x))^n*n))

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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*}

Mathematica [A]  time = 0.0116902, 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.

[In]

Integrate[(-E^(-x) + E^x)^n,x]

[Out]

((-E^(-x) + E^x)^n*(-1 + E^(2*x))*Hypergeometric2F1[1, 1 + n/2, 1 - n/2, E^(2*x)])/n

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Maple [F]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int \left ( - \left ({{\rm e}^{x}} \right ) ^{-1}+{{\rm e}^{x}} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1/exp(x)+exp(x))^n,x)

[Out]

int((-1/exp(x)+exp(x))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-e^{\left (-x\right )} + e^{x}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1/exp(x)+exp(x))^n,x, algorithm="maxima")

[Out]

integrate((-e^(-x) + e^x)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-e^{\left (-x\right )} + e^{x}\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1/exp(x)+exp(x))^n,x, algorithm="fricas")

[Out]

integral((-e^(-x) + e^x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e^{x} - e^{- x}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1/exp(x)+exp(x))**n,x)

[Out]

Integral((exp(x) - exp(-x))**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-e^{\left (-x\right )} + e^{x}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1/exp(x)+exp(x))^n,x, algorithm="giac")

[Out]

integrate((-e^(-x) + e^x)^n, x)