\(\int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx\) [23]

Optimal result

Integrand size = 29, antiderivative size = 114 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=-\frac {F^{c (a+b x)} \left (f^2-g^2\right )}{2 b c \log (F)}-\frac {e^{-2 d-2 e x} F^{c (a+b x)} (f-g)^2}{4 (2 e-b c \log (F))}+\frac {e^{2 d+2 e x} F^{c (a+b x)} (f+g)^2}{4 (2 e+b c \log (F))} \] Output:

-1/2*F^(c*(b*x+a))*(f^2-g^2)/b/c/ln(F)-exp(-2*e*x-2*d)*F^(c*(b*x+a))*(f-g) 
^2/(8*e-4*b*c*ln(F))+exp(2*e*x+2*d)*F^(c*(b*x+a))*(f+g)^2/(8*e+4*b*c*ln(F) 
)
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\frac {F^{c (a+b x)} \left (b c \cosh (2 (d+e x)) \log (F) \left (-4 e f g+b c \left (f^2+g^2\right ) \log (F)\right )+\left (f^2-g^2\right ) \left (4 e^2-b^2 c^2 \log ^2(F)\right )+2 b c \log (F) \left (-e \left (f^2+g^2\right )+b c f g \log (F)\right ) \sinh (2 (d+e x))\right )}{-8 b c e^2 \log (F)+2 b^3 c^3 \log ^3(F)} \] Input:

Integrate[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^2,x]
 

Output:

(F^(c*(a + b*x))*(b*c*Cosh[2*(d + e*x)]*Log[F]*(-4*e*f*g + b*c*(f^2 + g^2) 
*Log[F]) + (f^2 - g^2)*(4*e^2 - b^2*c^2*Log[F]^2) + 2*b*c*Log[F]*(-(e*(f^2 
 + g^2)) + b*c*f*g*Log[F])*Sinh[2*(d + e*x)]))/(-8*b*c*e^2*Log[F] + 2*b^3* 
c^3*Log[F]^3)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(378\) vs. \(2(114)=228\).

Time = 0.75 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.32, 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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^2,x]
 

Output:

(2*e*f*F^(a*c + b*c*x)*g*Cosh[2*d + 2*e*x])/(4*e^2 - b^2*c^2*Log[F]^2) - ( 
2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) + (2*e^ 
2*F^(a*c + b*c*x)*g^2)/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(a 
*c + b*c*x)*g^2*Cosh[d + e*x]^2*Log[F])/(4*e^2 - b^2*c^2*Log[F]^2) + (2*e* 
f^2*F^(a*c + b*c*x)*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2 
) + (2*e*F^(a*c + b*c*x)*g^2*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2 
*Log[F]^2) - (b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sinh[d + e*x]^2)/(4*e^2 - b^2 
*c^2*Log[F]^2) - (b*c*f*F^(a*c + b*c*x)*g*Log[F]*Sinh[2*d + 2*e*x])/(4*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. \(331\) vs. \(2(106)=212\).

Time = 67.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.91

method	result	size
risch	\(\frac {\left (\ln \left (F \right )^{2} b^{2} c^{2} f^{2} {\mathrm e}^{4 e x +4 d}+2 \ln \left (F \right )^{2} b^{2} c^{2} f g \,{\mathrm e}^{4 e x +4 d}+\ln \left (F \right )^{2} b^{2} c^{2} g^{2} {\mathrm e}^{4 e x +4 d}-2 \ln \left (F \right )^{2} b^{2} c^{2} f^{2} {\mathrm e}^{2 e x +2 d}+2 \ln \left (F \right )^{2} b^{2} c^{2} g^{2} {\mathrm e}^{2 e x +2 d}-2 \ln \left (F \right ) b c e \,f^{2} {\mathrm e}^{4 e x +4 d}-4 \ln \left (F \right ) b c e f g \,{\mathrm e}^{4 e x +4 d}-2 \ln \left (F \right ) b c e \,g^{2} {\mathrm e}^{4 e x +4 d}+\ln \left (F \right )^{2} b^{2} c^{2} f^{2}-2 \ln \left (F \right )^{2} b^{2} c^{2} f g +\ln \left (F \right )^{2} b^{2} c^{2} g^{2}+2 \ln \left (F \right ) b c e \,f^{2}-4 \ln \left (F \right ) b c e f g +2 \ln \left (F \right ) b c e \,g^{2}+8 e^{2} f^{2} {\mathrm e}^{2 e x +2 d}-8 e^{2} g^{2} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 b c \ln \left (F \right ) \left (b c \ln \left (F \right )-2 e \right ) \left (2 e +b c \ln \left (F \right )\right )}\)	\(332\)
orering	\(\frac {\left (3 b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right ) F^{c \left (b x +a \right )} \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{2}}{b c \ln \left (F \right ) \left (b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right )}-\frac {3 \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{2}+2 F^{c \left (b x +a \right )} \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right ) \left (g e \sinh \left (e x +d \right )+f e \cosh \left (e x +d \right )\right )\right )}{b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}}+\frac {F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{2}+4 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right ) \left (g e \sinh \left (e x +d \right )+f e \cosh \left (e x +d \right )\right )+2 F^{c \left (b x +a \right )} \left (g e \sinh \left (e x +d \right )+f e \cosh \left (e x +d \right )\right )^{2}+2 F^{c \left (b x +a \right )} \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right ) \left (g \,e^{2} \cosh \left (e x +d \right )+f \,e^{2} \sinh \left (e x +d \right )\right )}{b c \ln \left (F \right ) \left (b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right )}\)	\(384\)

Input:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*(ln(F)^2*b^2*c^2*f^2*exp(4*e*x+4*d)+2*ln(F)^2*b^2*c^2*f*g*exp(4*e*x+4* 
d)+ln(F)^2*b^2*c^2*g^2*exp(4*e*x+4*d)-2*ln(F)^2*b^2*c^2*f^2*exp(2*e*x+2*d) 
+2*ln(F)^2*b^2*c^2*g^2*exp(2*e*x+2*d)-2*ln(F)*b*c*e*f^2*exp(4*e*x+4*d)-4*l 
n(F)*b*c*e*f*g*exp(4*e*x+4*d)-2*ln(F)*b*c*e*g^2*exp(4*e*x+4*d)+ln(F)^2*b^2 
*c^2*f^2-2*ln(F)^2*b^2*c^2*f*g+ln(F)^2*b^2*c^2*g^2+2*ln(F)*b*c*e*f^2-4*ln( 
F)*b*c*e*f*g+2*ln(F)*b*c*e*g^2+8*e^2*f^2*exp(2*e*x+2*d)-8*e^2*g^2*exp(2*e* 
x+2*d))/b/c/ln(F)/(b*c*ln(F)-2*e)*exp(-2*e*x-2*d)/(2*e+b*c*ln(F))*F^(c*(b* 
x+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1387 vs. \(2 (105) = 210\).

Time = 0.10 (sec) , antiderivative size = 1387, normalized size of antiderivative = 12.17 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x, algorithm="fric 
as")
 

Output:

1/4*((((b^2*c^2*f^2 + 2*b^2*c^2*f*g + b^2*c^2*g^2)*log(F)^2 - 2*(b*c*e*f^2 
 + 2*b*c*e*f*g + b*c*e*g^2)*log(F))*sinh(e*x + d)^4 + 4*((b^2*c^2*f^2 + 2* 
b^2*c^2*f*g + b^2*c^2*g^2)*cosh(e*x + d)*log(F)^2 - 2*(b*c*e*f^2 + 2*b*c*e 
*f*g + b*c*e*g^2)*cosh(e*x + d)*log(F))*sinh(e*x + d)^3 + 8*(e^2*f^2 - e^2 
*g^2)*cosh(e*x + d)^2 + (b^2*c^2*f^2 - 2*b^2*c^2*f*g + b^2*c^2*g^2 + (b^2* 
c^2*f^2 + 2*b^2*c^2*f*g + b^2*c^2*g^2)*cosh(e*x + d)^4 - 2*(b^2*c^2*f^2 - 
b^2*c^2*g^2)*cosh(e*x + d)^2)*log(F)^2 + 2*(4*e^2*f^2 - 4*e^2*g^2 - 6*(b*c 
*e*f^2 + 2*b*c*e*f*g + b*c*e*g^2)*cosh(e*x + d)^2*log(F) - (b^2*c^2*f^2 - 
b^2*c^2*g^2 - 3*(b^2*c^2*f^2 + 2*b^2*c^2*f*g + b^2*c^2*g^2)*cosh(e*x + d)^ 
2)*log(F)^2)*sinh(e*x + d)^2 + 2*(b*c*e*f^2 - 2*b*c*e*f*g + b*c*e*g^2 - (b 
*c*e*f^2 + 2*b*c*e*f*g + b*c*e*g^2)*cosh(e*x + d)^4)*log(F) - 4*(2*(b*c*e* 
f^2 + 2*b*c*e*f*g + b*c*e*g^2)*cosh(e*x + d)^3*log(F) - ((b^2*c^2*f^2 + 2* 
b^2*c^2*f*g + b^2*c^2*g^2)*cosh(e*x + d)^3 - (b^2*c^2*f^2 - b^2*c^2*g^2)*c 
osh(e*x + d))*log(F)^2 - 4*(e^2*f^2 - e^2*g^2)*cosh(e*x + d))*sinh(e*x + d 
))*cosh((b*c*x + a*c)*log(F)) + (((b^2*c^2*f^2 + 2*b^2*c^2*f*g + b^2*c^2*g 
^2)*log(F)^2 - 2*(b*c*e*f^2 + 2*b*c*e*f*g + b*c*e*g^2)*log(F))*sinh(e*x + 
d)^4 + 4*((b^2*c^2*f^2 + 2*b^2*c^2*f*g + b^2*c^2*g^2)*cosh(e*x + d)*log(F) 
^2 - 2*(b*c*e*f^2 + 2*b*c*e*f*g + b*c*e*g^2)*cosh(e*x + d)*log(F))*sinh(e* 
x + d)^3 + 8*(e^2*f^2 - e^2*g^2)*cosh(e*x + d)^2 + (b^2*c^2*f^2 - 2*b^2*c^ 
2*f*g + b^2*c^2*g^2 + (b^2*c^2*f^2 + 2*b^2*c^2*f*g + b^2*c^2*g^2)*cosh(...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2082 vs. \(2 (95) = 190\).

Time = 1.50 (sec) , antiderivative size = 2082, normalized size of antiderivative = 18.26 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\text {Too large to display} \] Input:

integrate(F**(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))**2,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (105) = 210\).

Time = 0.06 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.37 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\frac {1}{4} \, g^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{b c \log \left (F\right ) + 2 \, e} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}} + \frac {2 \, F^{b c x + a c}}{b c \log \left (F\right )}\right )} + \frac {1}{4} \, f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{b c \log \left (F\right ) + 2 \, e} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}} - \frac {2 \, F^{b c x + a c}}{b c \log \left (F\right )}\right )} + \frac {1}{2} \, f g {\left (\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{b c \log \left (F\right ) + 2 \, e} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}}\right )} \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x, algorithm="maxi 
ma")
 

Output:

1/4*g^2*(F^(a*c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) + F^(a* 
c)*e^(b*c*x*log(F) - 2*e*x)/(b*c*e^(2*d)*log(F) - 2*e*e^(2*d)) + 2*F^(b*c* 
x + a*c)/(b*c*log(F))) + 1/4*f^2*(F^(a*c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/( 
b*c*log(F) + 2*e) + F^(a*c)*e^(b*c*x*log(F) - 2*e*x)/(b*c*e^(2*d)*log(F) - 
 2*e*e^(2*d)) - 2*F^(b*c*x + a*c)/(b*c*log(F))) + 1/2*f*g*(F^(a*c)*e^(b*c* 
x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) - F^(a*c)*e^(b*c*x*log(F) - 2*e 
*x)/(b*c*e^(2*d)*log(F) - 2*e*e^(2*d)))
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 1031, normalized size of antiderivative = 9.04 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x, algorithm="giac 
")
 

Output:

-(2*(f^2 - g^2)*b*c*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*s 
gn(F) + 1/2*pi*a*c)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) 
- pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*(f^2 - g^2)*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)/(4*b^2*c^2*log(abs(F)) 
^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) 
- I*((I*f^2 - I*g^2)*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)/(2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs 
(F))) - (I*f^2 - I*g^2)*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)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*lo 
g(abs(F))))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 1/2*(2*(b*c*log(abs( 
F)) + 2*e)*(f^2 + 2*f*g + g^2)*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(ab 
s(F)) + 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*(f^2 + 2*f*g + g^2)*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*sg 
n(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 2*e)^2))*e^(a*c*log(abs(F)) + (b*c 
*log(abs(F)) + 2*e)*x + 2*d) - I*((-I*f^2 - 2*I*f*g - I*g^2)*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)/(4*I*pi 
*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) + 16*e) - (-I*f^2 - 2*I*f*g - 
 I*g^2)*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)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) + 1...
 

Mupad [B] (verification not implemented)

Time = 3.56 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.18 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\frac {F^{b\,c\,x}\,F^{a\,c}\,\mathrm {cosh}\left (d+e\,x\right )\,\mathrm {sinh}\left (d+e\,x\right )\,\left (e\,\left (2\,f^2+2\,g^2\right )-2\,b\,c\,f\,g\,\ln \left (F\right )\right )}{4\,e^2-b^2\,c^2\,{\ln \left (F\right )}^2}-\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\left (b^2\,c^2\,g^2\,{\ln \left (F\right )}^2-2\,b\,c\,e\,f\,g\,\ln \left (F\right )+2\,e^2\,f^2-2\,e^2\,g^2\right )}{b\,c\,\ln \left (F\right )\,\left (4\,e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )}+\frac {F^{b\,c\,x}\,F^{a\,c}\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,\left (-b^2\,c^2\,f^2\,{\ln \left (F\right )}^2+2\,b\,c\,e\,f\,g\,\ln \left (F\right )+2\,e^2\,f^2-2\,e^2\,g^2\right )}{b\,c\,\ln \left (F\right )\,\left (4\,e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )} \] Input:

int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^2,x)
 

Output:

(F^(b*c*x)*F^(a*c)*cosh(d + e*x)*sinh(d + e*x)*(e*(2*f^2 + 2*g^2) - 2*b*c* 
f*g*log(F)))/(4*e^2 - b^2*c^2*log(F)^2) - (F^(b*c*x)*F^(a*c)*cosh(d + e*x) 
^2*(2*e^2*f^2 - 2*e^2*g^2 + b^2*c^2*g^2*log(F)^2 - 2*b*c*e*f*g*log(F)))/(b 
*c*log(F)*(4*e^2 - b^2*c^2*log(F)^2)) + (F^(b*c*x)*F^(a*c)*sinh(d + e*x)^2 
*(2*e^2*f^2 - 2*e^2*g^2 - b^2*c^2*f^2*log(F)^2 + 2*b*c*e*f*g*log(F)))/(b*c 
*log(F)*(4*e^2 - b^2*c^2*log(F)^2))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.22 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^2 \, dx=\frac {f^{b c x +a c} \left (\cosh \left (e x +d \right )^{2} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} g^{2}-2 \cosh \left (e x +d \right )^{2} \mathrm {log}\left (f \right ) b c e f g +2 \cosh \left (e x +d \right )^{2} e^{2} f^{2}-2 \cosh \left (e x +d \right )^{2} e^{2} g^{2}+2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right )^{2} \sinh \left (e x +d \right ) b^{2} c^{2} f g -2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c e \,f^{2}-2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c e \,g^{2}+\mathrm {log}\left (f \right )^{2} \sinh \left (e x +d \right )^{2} b^{2} c^{2} f^{2}-2 \,\mathrm {log}\left (f \right ) \sinh \left (e x +d \right )^{2} b c e f g -2 \sinh \left (e x +d \right )^{2} e^{2} f^{2}+2 \sinh \left (e x +d \right )^{2} e^{2} g^{2}\right )}{\mathrm {log}\left (f \right ) b c \left (\mathrm {log}\left (f \right )^{2} b^{2} c^{2}-4 e^{2}\right )} \] Input:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x)
 

Output:

(f**(a*c + b*c*x)*(cosh(d + e*x)**2*log(f)**2*b**2*c**2*g**2 - 2*cosh(d + 
e*x)**2*log(f)*b*c*e*f*g + 2*cosh(d + e*x)**2*e**2*f**2 - 2*cosh(d + e*x)* 
*2*e**2*g**2 + 2*cosh(d + e*x)*log(f)**2*sinh(d + e*x)*b**2*c**2*f*g - 2*c 
osh(d + e*x)*log(f)*sinh(d + e*x)*b*c*e*f**2 - 2*cosh(d + e*x)*log(f)*sinh 
(d + e*x)*b*c*e*g**2 + log(f)**2*sinh(d + e*x)**2*b**2*c**2*f**2 - 2*log(f 
)*sinh(d + e*x)**2*b*c*e*f*g - 2*sinh(d + e*x)**2*e**2*f**2 + 2*sinh(d + e 
*x)**2*e**2*g**2))/(log(f)*b*c*(log(f)**2*b**2*c**2 - 4*e**2))