\(\int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx\) [96]

Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\frac {e^{-d-e x} F^{c (a+b x)}}{8 (e-b c \log (F))}-\frac {e^{-3 d-3 e x} F^{c (a+b x)}}{8 (3 e-b c \log (F))}-\frac {e^{d+e x} F^{c (a+b x)}}{8 (e+b c \log (F))}+\frac {e^{3 d+3 e x} F^{c (a+b x)}}{8 (3 e+b c \log (F))} \] Output:

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

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\frac {1}{4} F^{c (a+b x)} \left (\frac {b c \cosh (d+e x) \log (F)-e \sinh (d+e x)}{(e-b c \log (F)) (e+b c \log (F))}+\frac {-b c \cosh (3 (d+e x)) \log (F)+3 e \sinh (3 (d+e x))}{9 e^2-b^2 c^2 \log ^2(F)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6035, 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.

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

\(\Big \downarrow \) 6035

\(\displaystyle \int \left (\frac {1}{4} \cosh (3 d+3 e x) F^{c (a+b x)}-\frac {1}{4} \cosh (d+e x) F^{c (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b*c*F^(c*(a + b*x))*Cosh[d + e*x]*Log[F])/(4*(e^2 - b^2*c^2*Log[F]^2)) - 
(b*c*F^(c*(a + b*x))*Cosh[3*d + 3*e*x]*Log[F])/(4*(9*e^2 - b^2*c^2*Log[F]^ 
2)) - (e*F^(c*(a + b*x))*Sinh[d + e*x])/(4*(e^2 - b^2*c^2*Log[F]^2)) + (3* 
e*F^(c*(a + b*x))*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 6035

Int[Cosh[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[( 
d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)) 
, Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g} 
, x] && IGtQ[m, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(124)=248\).

Time = 96.04 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.46

method	result	size
risch	\(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{6 e x +6 d}-\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{4 e x +4 d}-3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{6 e x +6 d}-\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{2 e x +2 d}+\ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{4 e x +4 d}-\ln \left (F \right ) b c \,e^{2} {\mathrm e}^{6 e x +6 d}+c^{3} b^{3} \ln \left (F \right )^{3}-\ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{2 e x +2 d}+9 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{4 e x +4 d}+3 e^{3} {\mathrm e}^{6 e x +6 d}+3 c^{2} b^{2} \ln \left (F \right )^{2} e +9 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{2 e x +2 d}-9 e^{3} {\mathrm e}^{4 e x +4 d}-e^{2} b c \ln \left (F \right )+9 e^{3} {\mathrm e}^{2 e x +2 d}-3 e^{3}\right ) {\mathrm e}^{-3 e x -3 d} F^{c \left (b x +a \right )}}{8 \left (b c \ln \left (F \right )-e \right ) \left (b c \ln \left (F \right )-3 e \right ) \left (e +b c \ln \left (F \right )\right ) \left (b c \ln \left (F \right )+3 e \right )}\)	\(325\)
orering	\(\frac {4 b c \ln \left (F \right ) \left (b^{2} c^{2} \ln \left (F \right )^{2}-5 e^{2}\right ) F^{c \left (b x +a \right )} \cosh \left (e x +d \right ) \sinh \left (e x +d \right )^{2}}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}-\frac {2 \left (3 b^{2} c^{2} \ln \left (F \right )^{2}-5 e^{2}\right ) \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right ) \sinh \left (e x +d \right )^{2}+F^{c \left (b x +a \right )} e \sinh \left (e x +d \right )^{3}+2 F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{2} \sinh \left (e x +d \right ) e \right )}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}+\frac {4 b c \ln \left (F \right ) \left (F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cosh \left (e x +d \right ) \sinh \left (e x +d \right )^{2}+2 F^{c \left (b x +a \right )} b c \ln \left (F \right ) e \sinh \left (e x +d \right )^{3}+4 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right )^{2} \sinh \left (e x +d \right ) e +7 F^{c \left (b x +a \right )} e^{2} \sinh \left (e x +d \right )^{2} \cosh \left (e x +d \right )+2 F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{3} e^{2}\right )}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}-\frac {F^{c \left (b x +a \right )} b^{3} c^{3} \ln \left (F \right )^{3} \cosh \left (e x +d \right ) \sinh \left (e x +d \right )^{2}+3 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} e \sinh \left (e x +d \right )^{3}+6 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cosh \left (e x +d \right )^{2} \sinh \left (e x +d \right ) e +21 F^{c \left (b x +a \right )} b c \ln \left (F \right ) e^{2} \sinh \left (e x +d \right )^{2} \cosh \left (e x +d \right )+6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right )^{3} e^{2}+20 F^{c \left (b x +a \right )} e^{3} \sinh \left (e x +d \right ) \cosh \left (e x +d \right )^{2}+7 F^{c \left (b x +a \right )} e^{3} \sinh \left (e x +d \right )^{3}}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}\)	\(634\)

Input:

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

Output:

1/8*(ln(F)^3*b^3*c^3*exp(6*e*x+6*d)-ln(F)^3*b^3*c^3*exp(4*e*x+4*d)-3*ln(F) 
^2*b^2*c^2*e*exp(6*e*x+6*d)-ln(F)^3*b^3*c^3*exp(2*e*x+2*d)+ln(F)^2*b^2*c^2 
*e*exp(4*e*x+4*d)-ln(F)*b*c*e^2*exp(6*e*x+6*d)+c^3*b^3*ln(F)^3-ln(F)^2*b^2 
*c^2*e*exp(2*e*x+2*d)+9*ln(F)*b*c*e^2*exp(4*e*x+4*d)+3*e^3*exp(6*e*x+6*d)+ 
3*c^2*b^2*ln(F)^2*e+9*ln(F)*b*c*e^2*exp(2*e*x+2*d)-9*e^3*exp(4*e*x+4*d)-e^ 
2*b*c*ln(F)+9*e^3*exp(2*e*x+2*d)-3*e^3)/(b*c*ln(F)-e)*exp(-3*e*x-3*d)/(b*c 
*ln(F)-3*e)/(e+b*c*ln(F))/(b*c*ln(F)+3*e)*F^(c*(b*x+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2224 vs. \(2 (120) = 240\).

Time = 0.16 (sec) , antiderivative size = 2224, normalized size of antiderivative = 16.85 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((3*e^3*cosh(e*x + d)^6 - 9*e^3*cosh(e*x + d)^4 + (b^3*c^3*log(F)^3 - 
3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c^ 
3*cosh(e*x + d)*log(F)^3 - 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2*co 
sh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 + 9*e^3*cosh(e*x 
 + d)^2 + (45*e^3*cosh(e*x + d)^2 + (15*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3) 
*log(F)^3 - 9*e^3 - (45*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - 
3*(5*b*c*e^2*cosh(e*x + d)^2 - 3*b*c*e^2)*log(F))*sinh(e*x + d)^4 + (b^3*c 
^3*cosh(e*x + d)^6 - b^3*c^3*cosh(e*x + d)^4 - b^3*c^3*cosh(e*x + d)^2 + b 
^3*c^3)*log(F)^3 + 4*(15*e^3*cosh(e*x + d)^3 - 9*e^3*cosh(e*x + d) + (5*b^ 
3*c^3*cosh(e*x + d)^3 - b^3*c^3*cosh(e*x + d))*log(F)^3 - (15*b^2*c^2*e*co 
sh(e*x + d)^3 - b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*cosh(e*x + 
d)^3 - 9*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3 - 3*e^3 - (3*b^2*c 
^2*e*cosh(e*x + d)^6 - b^2*c^2*e*cosh(e*x + d)^4 + b^2*c^2*e*cosh(e*x + d) 
^2 - 3*b^2*c^2*e)*log(F)^2 + (45*e^3*cosh(e*x + d)^4 - 54*e^3*cosh(e*x + d 
)^2 + (15*b^3*c^3*cosh(e*x + d)^4 - 6*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*l 
og(F)^3 + 9*e^3 - (45*b^2*c^2*e*cosh(e*x + d)^4 - 6*b^2*c^2*e*cosh(e*x + d 
)^2 + b^2*c^2*e)*log(F)^2 - 3*(5*b*c*e^2*cosh(e*x + d)^4 - 18*b*c*e^2*cosh 
(e*x + d)^2 - 3*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c*e^2*cosh(e*x + d)^ 
6 - 9*b*c*e^2*cosh(e*x + d)^4 - 9*b*c*e^2*cosh(e*x + d)^2 + b*c*e^2)*log(F 
) + 2*(9*e^3*cosh(e*x + d)^5 - 18*e^3*cosh(e*x + d)^3 + 9*e^3*cosh(e*x ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1476 vs. \(2 (117) = 234\).

Time = 3.25 (sec) , antiderivative size = 1476, normalized size of antiderivative = 11.18 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 3 \, e x + 3 \, d\right )}}{8 \, {\left (b c \log \left (F\right ) + 3 \, e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{8 \, {\left (b c \log \left (F\right ) + e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{8 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 3 \, e x\right )}}{8 \, {\left (b c e^{\left (3 \, d\right )} \log \left (F\right ) - 3 \, e e^{\left (3 \, d\right )}\right )}} \] Input:

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

Output:

1/8*F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) - 1/8*F^(a*c 
)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) - 1/8*F^(a*c)*e^(b*c*x*log(F 
) - e*x)/(b*c*e^d*log(F) - e*e^d) + 1/8*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 = 0.16 (sec) , antiderivative size = 1211, normalized size of antiderivative = 9.17 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(2*(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*sgn(F) - pi*b*c)^2 + 4*(b*c*log(ab 
s(F)) + 3*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)) + 3*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)* 
x + 3*d) + 1/2*I*(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)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs( 
F)) + 24*e) - 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)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F) 
) + 24*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 1/4*(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(a 
bs(F)) + e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 1/2*I* 
(-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)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) + 8*e) + 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)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) + 8*e))*e^(a*c* 
log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) - 1/4*(2*(b*c*log(abs(F)) - ...
 

Mupad [B] (verification not implemented)

Time = 3.52 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.30 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^3\,c^3\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,{\ln \left (F\right )}^3-2\,b^2\,c^2\,e\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \left (F\right )}^2-b^2\,c^2\,e\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,{\ln \left (F\right )}^2+2\,b\,c\,e^2\,{\mathrm {cosh}\left (d+e\,x\right )}^3\,\ln \left (F\right )-3\,b\,c\,e^2\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,\ln \left (F\right )+3\,e^3\,{\mathrm {sinh}\left (d+e\,x\right )}^3\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4} \] Input:

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

Output:

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

Reduce [F]

\[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cosh \left (e x +d \right ) \sinh \left (e x +d \right )^{2}d x \right ) \] Input:

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

Output:

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