3.9.84 \(\int F^{c (a+b x)} \cosh ^n(d+e x) \, dx\) [884]

Optimal. Leaf size=95 \[ -\frac {\left (1+e^{2 (d+e x)}\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \, _2F_1\left (-n,-\frac {e n-b c \log (F)}{2 e};\frac {1}{2} \left (2-n+\frac {b c \log (F)}{e}\right );-e^{2 (d+e x)}\right )}{e n-b c \log (F)} \]

[Out]

-F^(c*(b*x+a))*cosh(e*x+d)^n*hypergeom([-n, 1/2*(-e*n+b*c*ln(F))/e],[1-1/2*n+1/2*b*c*ln(F)/e],-exp(2*e*x+2*d))
/((1+exp(2*e*x+2*d))^n)/(e*n-b*c*ln(F))

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Rubi [A]
time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5591, 2291} \begin {gather*} -\frac {\left (e^{2 (d+e x)}+1\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \, _2F_1\left (-n,-\frac {e n-b c \log (F)}{2 e};\frac {1}{2} \left (-n+\frac {b c \log (F)}{e}+2\right );-e^{2 (d+e x)}\right )}{e n-b c \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Cosh[d + e*x]^n,x]

[Out]

-((F^(c*(a + b*x))*Cosh[d + e*x]^n*Hypergeometric2F1[-n, -1/2*(e*n - b*c*Log[F])/e, (2 - n + (b*c*Log[F])/e)/2
, -E^(2*(d + e*x))])/((1 + E^(2*(d + e*x)))^n*(e*n - b*c*Log[F])))

Rule 2291

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_)))*(H_)^((t_.)*((r_.)
+ (s_.)*(x_))), x_Symbol] :> Simp[G^(h*(f + g*x))*H^(t*(r + s*x))*((a + b*F^(e*(c + d*x)))^p/((g*h*Log[G] + s
*t*Log[H])*((a + b*F^(e*(c + d*x)))/a)^p))*Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]), (g*h*
Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, H, a, b, c, d, e,
f, g, h, r, s, t, p}, x] &&  !IntegerQ[p]

Rule 5591

Int[Cosh[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[E^(n*(d + e*x))*(Cosh[d
+ e*x]^n/(1 + E^(2*(d + e*x)))^n), Int[F^(c*(a + b*x))*((1 + E^(2*(d + e*x)))^n/E^(n*(d + e*x))), x], x] /; F
reeQ[{F, a, b, c, d, e, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx &=\left (e^{n (d+e x)} \left (1+e^{2 (d+e x)}\right )^{-n} \cosh ^n(d+e x)\right ) \int e^{-n (d+e x)} \left (1+e^{2 (d+e x)}\right )^n F^{c (a+b x)} \, dx\\ &=-\frac {\left (1+e^{2 (d+e x)}\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \, _2F_1\left (-n,-\frac {e n-b c \log (F)}{2 e};\frac {1}{2} \left (2-n+\frac {b c \log (F)}{e}\right );-e^{2 (d+e x)}\right )}{e n-b c \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 96, normalized size = 1.01 \begin {gather*} \frac {\left (1+e^{2 (d+e x)}\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \, _2F_1\left (-n,\frac {-e n+b c \log (F)}{2 e};1+\frac {-e n+b c \log (F)}{2 e};-e^{2 (d+e x)}\right )}{-e n+b c \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^n,x]

[Out]

(F^(c*(a + b*x))*Cosh[d + e*x]^n*Hypergeometric2F1[-n, (-(e*n) + b*c*Log[F])/(2*e), 1 + (-(e*n) + b*c*Log[F])/
(2*e), -E^(2*(d + e*x))])/((1 + E^(2*(d + e*x)))^n*(-(e*n) + b*c*Log[F]))

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Maple [F]
time = 0.77, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (\cosh ^{n}\left (e x +d \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*cosh(e*x+d)^n,x)

[Out]

int(F^(c*(b*x+a))*cosh(e*x+d)^n,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^n,x, algorithm="maxima")

[Out]

integrate(F^((b*x + a)*c)*cosh(x*e + d)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^n,x, algorithm="fricas")

[Out]

integral(F^(b*c*x + a*c)*cosh(x*e + d)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{c \left (a + b x\right )} \cosh ^{n}{\left (d + e x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*cosh(e*x+d)**n,x)

[Out]

Integral(F**(c*(a + b*x))*cosh(d + e*x)**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^n,x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)*cosh(e*x + d)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,{\mathrm {cosh}\left (d+e\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))*cosh(d + e*x)^n,x)

[Out]

int(F^(c*(a + b*x))*cosh(d + e*x)^n, x)

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