\(\int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx\) [22]

Optimal result

Integrand size = 29, antiderivative size = 164 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx=-\frac {3 e^{-d-e x} F^{c (a+b x)} (f-g)^2 (f+g)}{8 (e-b c \log (F))}+\frac {e^{-3 d-3 e x} F^{c (a+b x)} (f-g)^3}{8 (3 e-b c \log (F))}-\frac {3 e^{d+e x} F^{c (a+b x)} (f-g) (f+g)^2}{8 (e+b c \log (F))}+\frac {e^{3 d+3 e x} F^{c (a+b x)} (f+g)^3}{8 (3 e+b c \log (F))} \] Output:

-3*exp(-e*x-d)*F^(c*(b*x+a))*(f-g)^2*(f+g)/(8*e-8*b*c*ln(F))+exp(-3*e*x-3* 
d)*F^(c*(b*x+a))*(f-g)^3/(24*e-8*b*c*ln(F))-3*exp(e*x+d)*F^(c*(b*x+a))*(f- 
g)*(f+g)^2/(8*e+8*b*c*ln(F))+exp(3*e*x+3*d)*F^(c*(b*x+a))*(f+g)^3/(24*e+8* 
b*c*ln(F))
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.37 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx=\frac {1}{4} F^{c (a+b x)} \left (\frac {3 (-f+g) (f+g) \cosh (d+e x) (e f-b c g \log (F))}{(e-b c \log (F)) (e+b c \log (F))}+\frac {\cosh (3 (d+e x)) \left (3 e f \left (f^2+3 g^2\right )-b c g \left (3 f^2+g^2\right ) \log (F)\right )}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 (f-g) (f+g) (-e g+b c f \log (F)) \sinh (d+e x)}{(e-b c \log (F)) (e+b c \log (F))}+\frac {\left (3 e g \left (3 f^2+g^2\right )-b c f \left (f^2+3 g^2\right ) \log (F)\right ) \sinh (3 (d+e x))}{9 e^2-b^2 c^2 \log ^2(F)}\right ) \] Input:

Integrate[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^3,x]
 

Output:

(F^(c*(a + b*x))*((3*(-f + g)*(f + g)*Cosh[d + e*x]*(e*f - b*c*g*Log[F]))/ 
((e - b*c*Log[F])*(e + b*c*Log[F])) + (Cosh[3*(d + e*x)]*(3*e*f*(f^2 + 3*g 
^2) - b*c*g*(3*f^2 + g^2)*Log[F]))/(9*e^2 - b^2*c^2*Log[F]^2) + (3*(f - g) 
*(f + g)*(-(e*g) + b*c*f*Log[F])*Sinh[d + e*x])/((e - b*c*Log[F])*(e + b*c 
*Log[F])) + ((3*e*g*(3*f^2 + g^2) - b*c*f*(f^2 + 3*g^2)*Log[F])*Sinh[3*(d 
+ e*x)])/(9*e^2 - b^2*c^2*Log[F]^2)))/4
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(811\) vs. \(2(164)=328\).

Time = 1.41 (sec) , antiderivative size = 811, normalized size of antiderivative = 4.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {7292, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int F^{c (a+b x)} (f \sinh (d+e x)+g \cosh (d+e x))^3 \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int F^{a c+b c x} (f \sinh (d+e x)+g \cosh (d+e x))^3dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (f^3 \sinh ^3(d+e x) F^{a c+b c x}+3 f^2 g \sinh ^2(d+e x) \cosh (d+e x) F^{a c+b c x}+3 f g^2 \sinh (d+e x) \cosh ^2(d+e x) F^{a c+b c x}+g^3 \cosh ^3(d+e x) F^{a c+b c x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b c f^3 \log (F) \sinh ^3(d+e x) F^{a c+b x c}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e f^3 \cosh (d+e x) \sinh ^2(d+e x) F^{a c+b x c}}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {3 e f^2 g \sinh (d+e x) F^{a c+b x c}}{4 \left (e^2-b^2 c^2 \log ^2(F)\right )}-\frac {3 b c f g^2 \log (F) \sinh (d+e x) F^{a c+b x c}}{4 \left (e^2-b^2 c^2 \log ^2(F)\right )}+\frac {3 e g^3 \cosh ^2(d+e x) \sinh (d+e x) F^{a c+b x c}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {6 e^3 g^3 \sinh (d+e x) F^{a c+b x c}}{9 e^4-10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}+\frac {6 b c e^2 f^3 \log (F) \sinh (d+e x) F^{a c+b x c}}{9 e^4-10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}+\frac {9 e f^2 g \sinh (3 d+3 e x) F^{a c+b x c}}{4 \left (9 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {3 b c f g^2 \log (F) \sinh (3 d+3 e x) F^{a c+b x c}}{4 \left (9 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {3 e f g^2 \cosh (d+e x) F^{a c+b x c}}{4 \left (e^2-b^2 c^2 \log ^2(F)\right )}+\frac {3 b c f^2 g \cosh (d+e x) \log (F) F^{a c+b x c}}{4 \left (e^2-b^2 c^2 \log ^2(F)\right )}+\frac {9 e f g^2 \cosh (3 d+3 e x) F^{a c+b x c}}{4 \left (9 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c g^3 \cosh ^3(d+e x) \log (F) F^{a c+b x c}}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {3 b c f^2 g \cosh (3 d+3 e x) \log (F) F^{a c+b x c}}{4 \left (9 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {6 e^3 f^3 \cosh (d+e x) F^{a c+b x c}}{9 e^4-10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}-\frac {6 b c e^2 g^3 \cosh (d+e x) \log (F) F^{a c+b x c}}{9 e^4-10 b^2 c^2 \log ^2(F) e^2+b^4 c^4 \log ^4(F)}\)

Input:

Int[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^3,x]
 

Output:

(3*e*f*F^(a*c + b*c*x)*g^2*Cosh[d + e*x])/(4*(e^2 - b^2*c^2*Log[F]^2)) + ( 
3*b*c*f^2*F^(a*c + b*c*x)*g*Cosh[d + e*x]*Log[F])/(4*(e^2 - b^2*c^2*Log[F] 
^2)) + (9*e*f*F^(a*c + b*c*x)*g^2*Cosh[3*d + 3*e*x])/(4*(9*e^2 - b^2*c^2*L 
og[F]^2)) - (b*c*F^(a*c + b*c*x)*g^3*Cosh[d + e*x]^3*Log[F])/(9*e^2 - b^2* 
c^2*Log[F]^2) - (3*b*c*f^2*F^(a*c + b*c*x)*g*Cosh[3*d + 3*e*x]*Log[F])/(4* 
(9*e^2 - b^2*c^2*Log[F]^2)) - (6*e^3*f^3*F^(a*c + b*c*x)*Cosh[d + e*x])/(9 
*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) - (6*b*c*e^2*F^(a*c + b 
*c*x)*g^3*Cosh[d + e*x]*Log[F])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4 
*Log[F]^4) - (3*e*f^2*F^(a*c + b*c*x)*g*Sinh[d + e*x])/(4*(e^2 - b^2*c^2*L 
og[F]^2)) - (3*b*c*f*F^(a*c + b*c*x)*g^2*Log[F]*Sinh[d + e*x])/(4*(e^2 - b 
^2*c^2*Log[F]^2)) + (3*e*F^(a*c + b*c*x)*g^3*Cosh[d + e*x]^2*Sinh[d + e*x] 
)/(9*e^2 - b^2*c^2*Log[F]^2) + (6*e^3*F^(a*c + b*c*x)*g^3*Sinh[d + e*x])/( 
9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (6*b*c*e^2*f^3*F^(a* 
c + b*c*x)*Log[F]*Sinh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^ 
4*Log[F]^4) + (3*e*f^3*F^(a*c + b*c*x)*Cosh[d + e*x]*Sinh[d + e*x]^2)/(9*e 
^2 - b^2*c^2*Log[F]^2) - (b*c*f^3*F^(a*c + b*c*x)*Log[F]*Sinh[d + e*x]^3)/ 
(9*e^2 - b^2*c^2*Log[F]^2) + (9*e*f^2*F^(a*c + b*c*x)*g*Sinh[3*d + 3*e*x]) 
/(4*(9*e^2 - b^2*c^2*Log[F]^2)) - (3*b*c*f*F^(a*c + b*c*x)*g^2*Log[F]*Sinh 
[3*d + 3*e*x])/(4*(9*e^2 - b^2*c^2*Log[F]^2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1339\) vs. \(2(156)=312\).

Time = 1.79 (sec) , antiderivative size = 1340, normalized size of antiderivative = 8.17

\[\text {Expression too large to display}\]

Input:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^3,x)
 

Output:

1/8*(27*ln(F)*b*c*e^2*f^2*g*exp(2*e*x+2*d)+27*ln(F)*b*c*e^2*f*g^2*exp(2*e* 
x+2*d)-27*e^3*f^3*exp(2*e*x+2*d)-27*e^3*g^3*exp(2*e*x+2*d)-27*e^3*f^3*exp( 
4*e*x+4*d)+27*e^3*g^3*exp(4*e*x+4*d)+3*ln(F)^3*b^3*c^3*f*g^2*exp(6*e*x+6*d 
)-3*ln(F)^3*b^3*c^3*f^2*g*exp(4*e*x+4*d)+3*ln(F)^3*b^3*c^3*f*g^2*exp(4*e*x 
+4*d)-3*ln(F)^2*b^2*c^2*e*f^3*exp(6*e*x+6*d)-ln(F)^3*b^3*c^3*f^3+ln(F)^3*b 
^3*c^3*g^3-3*ln(F)*b*c*e^2*f^2*g*exp(6*e*x+6*d)-3*ln(F)*b*c*e^2*f*g^2*exp( 
6*e*x+6*d)-3*ln(F)^2*b^2*c^2*e*f^2*g*exp(2*e*x+2*d)-3*ln(F)^2*b^2*c^2*e*f* 
g^2*exp(2*e*x+2*d)+27*ln(F)*b*c*e^2*f^2*g*exp(4*e*x+4*d)-27*ln(F)*b*c*e^2* 
f*g^2*exp(4*e*x+4*d)-9*e^3*f^2*g-3*ln(F)^2*b^2*c^2*e*g^3*exp(6*e*x+6*d)+9* 
e^3*f*g^2-9*ln(F)^2*b^2*c^2*e*f^2*g*exp(6*e*x+6*d)-9*ln(F)^2*b^2*c^2*e*f*g 
^2*exp(6*e*x+6*d)+3*ln(F)^2*b^2*c^2*e*f^2*g*exp(4*e*x+4*d)-3*ln(F)^2*b^2*c 
^2*e*f*g^2*exp(4*e*x+4*d)+9*e^3*f*g^2*exp(6*e*x+6*d)-27*ln(F)*b*c*e^2*f^3* 
exp(2*e*x+2*d)-27*ln(F)*b*c*e^2*g^3*exp(2*e*x+2*d)+3*ln(F)^3*b^3*c^3*f^2*g 
*exp(6*e*x+6*d)+3*ln(F)^2*b^2*c^2*e*f^3*exp(2*e*x+2*d)+3*ln(F)^2*b^2*c^2*e 
*g^3*exp(2*e*x+2*d)+27*ln(F)*b*c*e^2*f^3*exp(4*e*x+4*d)-27*ln(F)*b*c*e^2*g 
^3*exp(4*e*x+4*d)-3*ln(F)^3*b^3*c^3*f^2*g*exp(2*e*x+2*d)-3*ln(F)^3*b^3*c^3 
*f*g^2*exp(2*e*x+2*d)+3*ln(F)^2*b^2*c^2*e*f^3*exp(4*e*x+4*d)-3*ln(F)^2*b^2 
*c^2*e*g^3*exp(4*e*x+4*d)-ln(F)*b*c*e^2*f^3*exp(6*e*x+6*d)-ln(F)*b*c*e^2*g 
^3*exp(6*e*x+6*d)+ln(F)^3*b^3*c^3*f^3*exp(6*e*x+6*d)+ln(F)^3*b^3*c^3*g^3*e 
xp(6*e*x+6*d)+9*ln(F)^2*b^2*c^2*e*f^2*g-9*ln(F)^2*b^2*c^2*e*f*g^2-3*ln(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6892 vs. \(2 (152) = 304\).

Time = 0.46 (sec) , antiderivative size = 6892, normalized size of antiderivative = 42.02 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx=\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^3,x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6615 vs. \(2 (148) = 296\).

Time = 3.94 (sec) , antiderivative size = 6615, normalized size of antiderivative = 40.34 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx=\text {Too large to display} \] Input:

integrate(F**(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))**3,x)
 

Output:

Piecewise((x*(f*sinh(d) + g*cosh(d))**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x 
*(f*sinh(d) + g*cosh(d))**3, Eq(b, 0) & Eq(e, 0)), (x*(f*sinh(d) + g*cosh( 
d))**3, Eq(c, 0) & Eq(e, 0)), (-3*F**(a*c + b*c*x)*f**3*x*sinh(b*c*x*log(F 
) - d)**3/8 + 3*F**(a*c + b*c*x)*f**3*x*sinh(b*c*x*log(F) - d)**2*cosh(b*c 
*x*log(F) - d)/8 + 3*F**(a*c + b*c*x)*f**3*x*sinh(b*c*x*log(F) - d)*cosh(b 
*c*x*log(F) - d)**2/8 - 3*F**(a*c + b*c*x)*f**3*x*cosh(b*c*x*log(F) - d)** 
3/8 - 3*F**(a*c + b*c*x)*f**2*g*x*sinh(b*c*x*log(F) - d)**3/8 + 3*F**(a*c 
+ b*c*x)*f**2*g*x*sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log(F) - d)/8 + 3*F 
**(a*c + b*c*x)*f**2*g*x*sinh(b*c*x*log(F) - d)*cosh(b*c*x*log(F) - d)**2/ 
8 - 3*F**(a*c + b*c*x)*f**2*g*x*cosh(b*c*x*log(F) - d)**3/8 + 3*F**(a*c + 
b*c*x)*f*g**2*x*sinh(b*c*x*log(F) - d)**3/8 - 3*F**(a*c + b*c*x)*f*g**2*x* 
sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log(F) - d)/8 - 3*F**(a*c + b*c*x)*f* 
g**2*x*sinh(b*c*x*log(F) - d)*cosh(b*c*x*log(F) - d)**2/8 + 3*F**(a*c + b* 
c*x)*f*g**2*x*cosh(b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*g**3*x*sinh 
(b*c*x*log(F) - d)**3/8 - 3*F**(a*c + b*c*x)*g**3*x*sinh(b*c*x*log(F) - d) 
**2*cosh(b*c*x*log(F) - d)/8 - 3*F**(a*c + b*c*x)*g**3*x*sinh(b*c*x*log(F) 
 - d)*cosh(b*c*x*log(F) - d)**2/8 + 3*F**(a*c + b*c*x)*g**3*x*cosh(b*c*x*l 
og(F) - d)**3/8 + F**(a*c + b*c*x)*f**3*sinh(b*c*x*log(F) - d)**3/(8*b*c*l 
og(F)) - 3*F**(a*c + b*c*x)*f**3*sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log( 
F) - d)/(4*b*c*log(F)) + 3*F**(a*c + b*c*x)*f**3*cosh(b*c*x*log(F) - d)...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (152) = 304\).

Time = 0.10 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.37 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx =\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^3,x, algorithm="maxi 
ma")
 

Output:

1/8*g^3*(F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) + 3*F^( 
a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) + 3*F^(a*c)*e^(b*c*x*log( 
F) - e*x)/(b*c*e^d*log(F) - e*e^d) + F^(a*c)*e^(b*c*x*log(F) - 3*e*x)/(b*c 
*e^(3*d)*log(F) - 3*e*e^(3*d))) + 3/8*f*g^2*(F^(a*c)*e^(b*c*x*log(F) + 3*e 
*x + 3*d)/(b*c*log(F) + 3*e) + F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*log 
(F) + e) - F^(a*c)*e^(b*c*x*log(F) - e*x)/(b*c*e^d*log(F) - e*e^d) - F^(a* 
c)*e^(b*c*x*log(F) - 3*e*x)/(b*c*e^(3*d)*log(F) - 3*e*e^(3*d))) + 3/8*f^2* 
g*(F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) - F^(a*c)*e^( 
b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) - F^(a*c)*e^(b*c*x*log(F) - e*x)/ 
(b*c*e^d*log(F) - e*e^d) + F^(a*c)*e^(b*c*x*log(F) - 3*e*x)/(b*c*e^(3*d)*l 
og(F) - 3*e*e^(3*d))) + 1/8*f^3*(F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b 
*c*log(F) + 3*e) - 3*F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) + 
 3*F^(a*c)*e^(b*c*x*log(F) - e*x)/(b*c*e^d*log(F) - e*e^d) - F^(a*c)*e^(b* 
c*x*log(F) - 3*e*x)/(b*c*e^(3*d)*log(F) - 3*e*e^(3*d)))
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 1549, normalized size of antiderivative = 9.45 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx=\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^3,x, algorithm="giac 
")
 

Output:

1/4*(2*(f^3 + 3*f^2*g + 3*f*g^2 + g^3)*(b*c*log(abs(F)) + 3*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)) + 3*e)^2) - (pi*b*c*sgn(F) - pi*b*c) 
*(f^3 + 3*f^2*g + 3*f*g^2 + g^3)*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)) + 3*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 
 I*((-I*f^3 - 3*I*f^2*g - 3*I*f*g^2 - I*g^3)*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)/(8*I*pi*b*c*sgn(F) - 8* 
I*pi*b*c + 16*b*c*log(abs(F)) + 48*e) - (-I*f^3 - 3*I*f^2*g - 3*I*f*g^2 - 
I*g^3)*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*e 
))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 3/4*(2*(f^3 + f 
^2*g - f*g^2 - g^3)*(b*c*log(abs(F)) + e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*p 
i*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)*(f^3 + f^2*g - f*g^2 - 
 g^3)*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) - 3*I*((I*f^3 + I*f^2*g - I*f*g^2 
- I*g^3)*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 1...
 

Mupad [B] (verification not implemented)

Time = 4.29 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.34 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx=\frac {F^{b\,c\,x}\,F^{a\,c}\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,\left (e^2\,\left (6\,b\,c\,g^3\,\ln \left (F\right )-9\,b\,c\,f^2\,g\,\ln \left (F\right )\right )-e\,\left (3\,b^2\,c^2\,f^3\,{\ln \left (F\right )}^2+6\,b^2\,c^2\,f\,g^2\,{\ln \left (F\right )}^2\right )+9\,e^3\,f^3+3\,b^3\,c^3\,f^2\,g\,{\ln \left (F\right )}^3\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4}-\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,\left (e^2\,\left (7\,b\,c\,f^3\,\ln \left (F\right )-6\,b\,c\,f\,g^2\,\ln \left (F\right )\right )-e^3\,\left (9\,f^2\,g-6\,g^3\right )-b^3\,c^3\,f^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,e\,f^2\,g\,{\ln \left (F\right )}^2\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4}-\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {cosh}\left (d+e\,x\right )}^3\,\left (e^2\,\left (7\,b\,c\,g^3\,\ln \left (F\right )-6\,b\,c\,f^2\,g\,\ln \left (F\right )\right )-e^3\,\left (9\,f\,g^2-6\,f^3\right )-b^3\,c^3\,g^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,e\,f\,g^2\,{\ln \left (F\right )}^2\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4}+\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,\left (e^2\,\left (6\,b\,c\,f^3\,\ln \left (F\right )-9\,b\,c\,f\,g^2\,\ln \left (F\right )\right )-e\,\left (6\,b^2\,c^2\,f^2\,g\,{\ln \left (F\right )}^2+3\,b^2\,c^2\,g^3\,{\ln \left (F\right )}^2\right )+9\,e^3\,g^3+3\,b^3\,c^3\,f\,g^2\,{\ln \left (F\right )}^3\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4} \] Input:

int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^3,x)
 

Output:

(F^(b*c*x)*F^(a*c)*cosh(d + e*x)*sinh(d + e*x)^2*(e^2*(6*b*c*g^3*log(F) - 
9*b*c*f^2*g*log(F)) - e*(3*b^2*c^2*f^3*log(F)^2 + 6*b^2*c^2*f*g^2*log(F)^2 
) + 9*e^3*f^3 + 3*b^3*c^3*f^2*g*log(F)^3))/(9*e^4 + b^4*c^4*log(F)^4 - 10* 
b^2*c^2*e^2*log(F)^2) - (F^(b*c*x)*F^(a*c)*sinh(d + e*x)^3*(e^2*(7*b*c*f^3 
*log(F) - 6*b*c*f*g^2*log(F)) - e^3*(9*f^2*g - 6*g^3) - b^3*c^3*f^3*log(F) 
^3 + 3*b^2*c^2*e*f^2*g*log(F)^2))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e 
^2*log(F)^2) - (F^(b*c*x)*F^(a*c)*cosh(d + e*x)^3*(e^2*(7*b*c*g^3*log(F) - 
 6*b*c*f^2*g*log(F)) - e^3*(9*f*g^2 - 6*f^3) - b^3*c^3*g^3*log(F)^3 + 3*b^ 
2*c^2*e*f*g^2*log(F)^2))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e^2*log(F) 
^2) + (F^(b*c*x)*F^(a*c)*cosh(d + e*x)^2*sinh(d + e*x)*(e^2*(6*b*c*f^3*log 
(F) - 9*b*c*f*g^2*log(F)) - e*(3*b^2*c^2*g^3*log(F)^2 + 6*b^2*c^2*f^2*g*lo 
g(F)^2) + 9*e^3*g^3 + 3*b^3*c^3*f*g^2*log(F)^3))/(9*e^4 + b^4*c^4*log(F)^4 
 - 10*b^2*c^2*e^2*log(F)^2)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.78 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^3 \, dx =\text {Too large to display} \] Input:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^3,x)
                                                                                    
                                                                                    
 

Output:

(f**(a*c + b*c*x)*(cosh(d + e*x)**3*log(f)**3*b**3*c**3*g**3 - 3*cosh(d + 
e*x)**3*log(f)**2*b**2*c**2*e*f*g**2 + 6*cosh(d + e*x)**3*log(f)*b*c*e**2* 
f**2*g - 7*cosh(d + e*x)**3*log(f)*b*c*e**2*g**3 - 6*cosh(d + e*x)**3*e**3 
*f**3 + 9*cosh(d + e*x)**3*e**3*f*g**2 + 3*cosh(d + e*x)**2*log(f)**3*sinh 
(d + e*x)*b**3*c**3*f*g**2 - 6*cosh(d + e*x)**2*log(f)**2*sinh(d + e*x)*b* 
*2*c**2*e*f**2*g - 3*cosh(d + e*x)**2*log(f)**2*sinh(d + e*x)*b**2*c**2*e* 
g**3 + 6*cosh(d + e*x)**2*log(f)*sinh(d + e*x)*b*c*e**2*f**3 - 9*cosh(d + 
e*x)**2*log(f)*sinh(d + e*x)*b*c*e**2*f*g**2 + 9*cosh(d + e*x)**2*sinh(d + 
 e*x)*e**3*g**3 + 3*cosh(d + e*x)*log(f)**3*sinh(d + e*x)**2*b**3*c**3*f** 
2*g - 3*cosh(d + e*x)*log(f)**2*sinh(d + e*x)**2*b**2*c**2*e*f**3 - 6*cosh 
(d + e*x)*log(f)**2*sinh(d + e*x)**2*b**2*c**2*e*f*g**2 - 9*cosh(d + e*x)* 
log(f)*sinh(d + e*x)**2*b*c*e**2*f**2*g + 6*cosh(d + e*x)*log(f)*sinh(d + 
e*x)**2*b*c*e**2*g**3 + 9*cosh(d + e*x)*sinh(d + e*x)**2*e**3*f**3 + log(f 
)**3*sinh(d + e*x)**3*b**3*c**3*f**3 - 3*log(f)**2*sinh(d + e*x)**3*b**2*c 
**2*e*f**2*g - 7*log(f)*sinh(d + e*x)**3*b*c*e**2*f**3 + 6*log(f)*sinh(d + 
 e*x)**3*b*c*e**2*f*g**2 + 9*sinh(d + e*x)**3*e**3*f**2*g - 6*sinh(d + e*x 
)**3*e**3*g**3))/(log(f)**4*b**4*c**4 - 10*log(f)**2*b**2*c**2*e**2 + 9*e* 
*4)