3.324 \(\int F^{c (a+b x)} \sinh (d+e x) \, dx\)

Optimal. Leaf size=75 \[ \frac{e \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac{b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]

[Out]

(e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x])/(e^2 -
b^2*c^2*Log[F]^2)

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Rubi [A]  time = 0.0186574, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5474} \[ \frac{e \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac{b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Sinh[d + e*x],x]

[Out]

(e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x])/(e^2 -
b^2*c^2*Log[F]^2)

Rule 5474

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)
, x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.105476, size = 50, normalized size = 0.67 \[ \frac{F^{c (a+b x)} (e \cosh (d+e x)-b c \log (F) \sinh (d+e x))}{(e-b c \log (F)) (b c \log (F)+e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 77, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( F \right ) bc{{\rm e}^{2\,ex+2\,d}}-bc\ln \left ( F \right ) -e{{\rm e}^{2\,ex+2\,d}}-e \right ){{\rm e}^{-ex-d}}{F}^{c \left ( bx+a \right ) }}{ \left ( 2\,bc\ln \left ( F \right ) -2\,e \right ) \left ( e+bc\ln \left ( F \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(ln(F)*b*c*exp(2*e*x+2*d)-b*c*ln(F)-e*exp(2*e*x+2*d)-e)/(b*c*ln(F)-e)*exp(-e*x-d)/(e+b*c*ln(F))*F^(c*(b*x+
a))

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Maxima [A]  time = 1.07309, size = 85, normalized size = 1.13 \begin{align*} \frac{F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{2 \,{\left (b c \log \left (F\right ) + e\right )}} - \frac{F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{2 \,{\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) - 1/2*F^(a*c)*e^(b*c*x*log(F) - e*x)/(b*c*e^d*log(F) -
e*e^d)

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Fricas [B]  time = 1.83603, size = 643, normalized size = 8.57 \begin{align*} -\frac{{\left (e \cosh \left (e x + d\right )^{2} -{\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} -{\left (b c \cosh \left (e x + d\right )^{2} - b c\right )} \log \left (F\right ) - 2 \,{\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) + e\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) +{\left (e \cosh \left (e x + d\right )^{2} -{\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} -{\left (b c \cosh \left (e x + d\right )^{2} - b c\right )} \log \left (F\right ) - 2 \,{\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) + e\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \,{\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - e^{2} \cosh \left (e x + d\right ) +{\left (b^{2} c^{2} \log \left (F\right )^{2} - e^{2}\right )} \sinh \left (e x + d\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((e*cosh(e*x + d)^2 - (b*c*log(F) - e)*sinh(e*x + d)^2 - (b*c*cosh(e*x + d)^2 - b*c)*log(F) - 2*(b*c*cosh
(e*x + d)*log(F) - e*cosh(e*x + d))*sinh(e*x + d) + e)*cosh((b*c*x + a*c)*log(F)) + (e*cosh(e*x + d)^2 - (b*c*
log(F) - e)*sinh(e*x + d)^2 - (b*c*cosh(e*x + d)^2 - b*c)*log(F) - 2*(b*c*cosh(e*x + d)*log(F) - e*cosh(e*x +
d))*sinh(e*x + d) + e)*sinh((b*c*x + a*c)*log(F)))/(b^2*c^2*cosh(e*x + d)*log(F)^2 - e^2*cosh(e*x + d) + (b^2*
c^2*log(F)^2 - e^2)*sinh(e*x + d))

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Sympy [A]  time = 13.8538, size = 316, normalized size = 4.21 \begin{align*} \begin{cases} \frac{\left (-1\right )^{a c} \left (-1\right )^{- \frac{i e x}{\pi }} x \sinh{\left (d + e x \right )}}{2} - \frac{\left (-1\right )^{a c} \left (-1\right )^{- \frac{i e x}{\pi }} x \cosh{\left (d + e x \right )}}{2} + \frac{\left (-1\right )^{a c} \left (-1\right )^{- \frac{i e x}{\pi }} \cosh{\left (d + e x \right )}}{2 e} & \text{for}\: F = -1 \wedge b = - \frac{i e}{\pi c} \\x \sinh{\left (d \right )} & \text{for}\: F = 1 \wedge e = 0 \\\tilde{\infty } e \left (e^{- \frac{e}{b c}}\right )^{a c} \left (e^{- \frac{e}{b c}}\right )^{b c x} \sinh{\left (d + e x \right )} + \tilde{\infty } e \left (e^{- \frac{e}{b c}}\right )^{a c} \left (e^{- \frac{e}{b c}}\right )^{b c x} \cosh{\left (d + e x \right )} & \text{for}\: F = e^{- \frac{e}{b c}} \\\tilde{\infty } e \left (e^{\frac{e}{b c}}\right )^{a c} \left (e^{\frac{e}{b c}}\right )^{b c x} \sinh{\left (d + e x \right )} + \tilde{\infty } e \left (e^{\frac{e}{b c}}\right )^{a c} \left (e^{\frac{e}{b c}}\right )^{b c x} \cosh{\left (d + e x \right )} & \text{for}\: F = e^{\frac{e}{b c}} \\\frac{F^{a c} F^{b c x} b c \log{\left (F \right )} \sinh{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} - e^{2}} - \frac{F^{a c} F^{b c x} e \cosh{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} - e^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-1)**(a*c)*(-1)**(-I*e*x/pi)*x*sinh(d + e*x)/2 - (-1)**(a*c)*(-1)**(-I*e*x/pi)*x*cosh(d + e*x)/2 +
(-1)**(a*c)*(-1)**(-I*e*x/pi)*cosh(d + e*x)/(2*e), Eq(F, -1) & Eq(b, -I*e/(pi*c))), (x*sinh(d), Eq(F, 1) & Eq
(e, 0)), (zoo*e*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*sinh(d + e*x) + zoo*e*exp(-e/(b*c))**(a*c)*exp(-e/
(b*c))**(b*c*x)*cosh(d + e*x), Eq(F, exp(-e/(b*c)))), (zoo*e*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*sinh(d
+ e*x) + zoo*e*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*cosh(d + e*x), Eq(F, exp(e/(b*c)))), (F**(a*c)*F**(b*
c*x)*b*c*log(F)*sinh(d + e*x)/(b**2*c**2*log(F)**2 - e**2) - F**(a*c)*F**(b*c*x)*e*cosh(d + e*x)/(b**2*c**2*lo
g(F)**2 - e**2), True))

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Giac [C]  time = 1.20335, size = 826, normalized size = 11.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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