### 3.1050 $$\int F^{a+b x} (\cosh (c+d x)+\sinh (c+d x))^n \, dx$$

Optimal. Leaf size=27 $\frac{F^{a+b x} \left (e^{c+d x}\right )^n}{b \log (F)+d n}$

[Out]

((E^(c + d*x))^n*F^(a + b*x))/(d*n + b*Log[F])

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Rubi [A]  time = 0.0910117, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.174, Rules used = {5648, 2281, 2287, 2194} $\frac{F^{a+b x} \left (e^{c+d x}\right )^n}{b \log (F)+d n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((E^(c + d*x))^n*F^(a + b*x))/(d*n + b*Log[F])

Rule 5648

Int[(u_.)*(Cosh[v_]*(a_.) + (b_.)*Sinh[v_])^(n_.), x_Symbol] :> Int[u*(a*E^((a*v)/b))^n, x] /; FreeQ[{a, b, n}
, x] && EqQ[a^2 - b^2, 0]

Rule 2281

Int[(u_.)*((a_.)*(F_)^(v_))^(n_), x_Symbol] :> Dist[(a*F^v)^n/F^(n*v), Int[u*F^(n*v), x], x] /; FreeQ[{F, a, n
}, x] &&  !IntegerQ[n]

Rule 2287

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0861154, size = 33, normalized size = 1.22 $\frac{F^{a+b x} (\sinh (c+d x)+\cosh (c+d x))^n}{b \log (F)+d n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(F^(a + b*x)*(Cosh[c + d*x] + Sinh[c + d*x])^n)/(d*n + b*Log[F])

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Maple [A]  time = 0.012, size = 34, normalized size = 1.3 \begin{align*}{\frac{{F}^{bx+a} \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) ^{n}}{dn+b\ln \left ( F \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(d*n+b*ln(F))*F^(b*x+a)*(cosh(d*x+c)+sinh(d*x+c))^n

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Maxima [A]  time = 1.05323, size = 38, normalized size = 1.41 \begin{align*} \frac{F^{a} e^{\left (d n x + b x \log \left (F\right ) + c n\right )}}{d n + b \log \left (F\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^a*e^(d*n*x + b*x*log(F) + c*n)/(d*n + b*log(F))

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Fricas [B]  time = 2.07509, size = 200, normalized size = 7.41 \begin{align*} \frac{{\left (\cosh \left (d n x + c n\right ) + \sinh \left (d n x + c n\right )\right )} \cosh \left ({\left (b x + a\right )} \log \left (F\right )\right ) +{\left (\cosh \left (d n x + c n\right ) + \sinh \left (d n x + c n\right )\right )} \sinh \left ({\left (b x + a\right )} \log \left (F\right )\right )}{d n + b \log \left (F\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((cosh(d*n*x + c*n) + sinh(d*n*x + c*n))*cosh((b*x + a)*log(F)) + (cosh(d*n*x + c*n) + sinh(d*n*x + c*n))*sinh
((b*x + a)*log(F)))/(d*n + b*log(F))

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Sympy [A]  time = 8.38276, size = 94, normalized size = 3.48 \begin{align*} \begin{cases} \frac{F^{a} F^{b x} \left (\sinh{\left (c + d x \right )} + \cosh{\left (c + d x \right )}\right )^{n}}{b \log{\left (F \right )} + d n} & \text{for}\: b \neq - \frac{d n}{\log{\left (F \right )}} \\F^{a} x \left (\sinh{\left (c + d x \right )} + \cosh{\left (c + d x \right )}\right )^{n} e^{- d n x} + \frac{F^{a} \left (\sinh{\left (c + d x \right )} + \cosh{\left (c + d x \right )}\right )^{n} e^{- d n x}}{d n} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((F**a*F**(b*x)*(sinh(c + d*x) + cosh(c + d*x))**n/(b*log(F) + d*n), Ne(b, -d*n/log(F))), (F**a*x*(si
nh(c + d*x) + cosh(c + d*x))**n*exp(-d*n*x) + F**a*(sinh(c + d*x) + cosh(c + d*x))**n*exp(-d*n*x)/(d*n), True)
)

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Giac [C]  time = 1.2019, size = 370, normalized size = 13.7 \begin{align*} 2 \,{\left (\frac{2 \,{\left (d n + b \log \left ({\left | F \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi b x \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi b x - \frac{1}{2} \, \pi a \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi a\right )}{{\left (\pi b \mathrm{sgn}\left (F\right ) - \pi b\right )}^{2} + 4 \,{\left (d n + b \log \left ({\left | F \right |}\right )\right )}^{2}} - \frac{{\left (\pi b \mathrm{sgn}\left (F\right ) - \pi b\right )} \sin \left (-\frac{1}{2} \, \pi b x \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi b x - \frac{1}{2} \, \pi a \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi a\right )}{{\left (\pi b \mathrm{sgn}\left (F\right ) - \pi b\right )}^{2} + 4 \,{\left (d n + b \log \left ({\left | F \right |}\right )\right )}^{2}}\right )} e^{\left (c n +{\left (d n + b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} - \frac{1}{2} i \,{\left (-\frac{2 i \, e^{\left (\frac{1}{2} i \, \pi b x \mathrm{sgn}\left (F\right ) - \frac{1}{2} i \, \pi b x + \frac{1}{2} i \, \pi a \mathrm{sgn}\left (F\right ) - \frac{1}{2} i \, \pi a\right )}}{i \, \pi b \mathrm{sgn}\left (F\right ) - i \, \pi b + 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )} + \frac{2 i \, e^{\left (-\frac{1}{2} i \, \pi b x \mathrm{sgn}\left (F\right ) + \frac{1}{2} i \, \pi b x - \frac{1}{2} i \, \pi a \mathrm{sgn}\left (F\right ) + \frac{1}{2} i \, \pi a\right )}}{-i \, \pi b \mathrm{sgn}\left (F\right ) + i \, \pi b + 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )}\right )} e^{\left (c n +{\left (d n + b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*(d*n + b*log(abs(F)))*cos(-1/2*pi*b*x*sgn(F) + 1/2*pi*b*x - 1/2*pi*a*sgn(F) + 1/2*pi*a)/((pi*b*sgn(F) - p
i*b)^2 + 4*(d*n + b*log(abs(F)))^2) - (pi*b*sgn(F) - pi*b)*sin(-1/2*pi*b*x*sgn(F) + 1/2*pi*b*x - 1/2*pi*a*sgn(
F) + 1/2*pi*a)/((pi*b*sgn(F) - pi*b)^2 + 4*(d*n + b*log(abs(F)))^2))*e^(c*n + (d*n + b*log(abs(F)))*x + a*log(
abs(F))) - 1/2*I*(-2*I*e^(1/2*I*pi*b*x*sgn(F) - 1/2*I*pi*b*x + 1/2*I*pi*a*sgn(F) - 1/2*I*pi*a)/(I*pi*b*sgn(F)
- I*pi*b + 2*d*n + 2*b*log(abs(F))) + 2*I*e^(-1/2*I*pi*b*x*sgn(F) + 1/2*I*pi*b*x - 1/2*I*pi*a*sgn(F) + 1/2*I*p
i*a)/(-I*pi*b*sgn(F) + I*pi*b + 2*d*n + 2*b*log(abs(F))))*e^(c*n + (d*n + b*log(abs(F)))*x + a*log(abs(F)))