Integrand size = 24, antiderivative size = 132 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\frac {e^{-d-e x} F^{c (a+b x)}}{8 (e-b c \log (F))}+\frac {e^{-3 d-3 e x} F^{c (a+b x)}}{8 (3 e-b c \log (F))}+\frac {e^{d+e x} F^{c (a+b x)}}{8 (e+b c \log (F))}+\frac {e^{3 d+3 e x} F^{c (a+b x)}}{8 (3 e+b c \log (F))} \] 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))
Time = 0.63 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\frac {1}{4} F^{c (a+b x)} \left (\frac {e \cosh (d+e x)-b c \log (F) \sinh (d+e x)}{(e-b c \log (F)) (e+b c \log (F))}+\frac {3 e \cosh (3 (d+e x))-b c \log (F) \sinh (3 (d+e x))}{9 e^2-b^2 c^2 \log ^2(F)}\right ) \] Input:
Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^2*Sinh[d + e*x],x]
Output:
(F^(c*(a + b*x))*((e*Cosh[d + e*x] - b*c*Log[F]*Sinh[d + e*x])/((e - b*c*L og[F])*(e + b*c*Log[F])) + (3*e*Cosh[3*(d + e*x)] - b*c*Log[F]*Sinh[3*(d + e*x)])/(9*e^2 - b^2*c^2*Log[F]^2)))/4
Time = 0.33 (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.
\(\displaystyle \int \sinh (d+e x) \cosh ^2(d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 6035 |
\(\displaystyle \int \left (\frac {1}{4} \sinh (d+e x) F^{c (a+b x)}+\frac {1}{4} \sinh (3 d+3 e x) F^{c (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{4 \left (e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c \log (F) \sinh (3 d+3 e x) F^{c (a+b x)}}{4 \left (9 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {e \cosh (d+e x) F^{c (a+b x)}}{4 \left (e^2-b^2 c^2 \log ^2(F)\right )}+\frac {3 e \cosh (3 d+3 e x) F^{c (a+b x)}}{4 \left (9 e^2-b^2 c^2 \log ^2(F)\right )}\) |
Input:
Int[F^(c*(a + b*x))*Cosh[d + e*x]^2*Sinh[d + e*x],x]
Output:
(e*F^(c*(a + b*x))*Cosh[d + e*x])/(4*(e^2 - b^2*c^2*Log[F]^2)) + (3*e*F^(c *(a + b*x))*Cosh[3*d + 3*e*x])/(4*(9*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c* (a + b*x))*Log[F]*Sinh[d + e*x])/(4*(e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c* (a + b*x))*Log[F]*Sinh[3*d + 3*e*x])/(4*(9*e^2 - b^2*c^2*Log[F]^2))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(122)=244\).
Time = 34.59 (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 )^{2} \sinh \left (e x +d \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 {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 )^{2} \sinh \left (e x +d \right )+2 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{3} 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 )^{2} \sinh \left (e x +d \right )+4 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+2 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right )^{3} e +7 F^{c \left (b x +a \right )} \sinh \left (e x +d \right ) e^{2} \cosh \left (e x +d \right )^{2}+2 F^{c \left (b x +a \right )} \sinh \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 )^{2} \sinh \left (e x +d \right )+6 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+3 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cosh \left (e x +d \right )^{3} e +21 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right ) e^{2} \cosh \left (e x +d \right )^{2}+6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{3} e^{2}+7 F^{c \left (b x +a \right )} e^{3} \cosh \left (e x +d \right )^{3}+20 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{2} e^{3} \cosh \left (e x +d \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}}\) | \(634\) |
Input:
int(F^(c*(b*x+a))*cosh(e*x+d)^2*sinh(e*x+d),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))
Leaf count of result is larger than twice the leaf count of optimal. 2216 vs. \(2 (120) = 240\).
Time = 0.18 (sec) , antiderivative size = 2216, normalized size of antiderivative = 16.79 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^2*sinh(e*x+d),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 ...
Leaf count of result is larger than twice the leaf count of optimal. 1476 vs. \(2 (117) = 234\).
Time = 3.42 (sec) , antiderivative size = 1476, normalized size of antiderivative = 11.18 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*cosh(e*x+d)**2*sinh(e*x+d),x)
Output:
Piecewise((x*sinh(d)*cosh(d)**2, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sinh(d) *cosh(d)**2, Eq(b, 0) & Eq(e, 0)), (x*sinh(d)*cosh(d)**2, 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*s inh(b*c*x*log(F) - d)*cosh(b*c*x*log(F) - d)**2/8 + F**(a*c + b*c*x)*x*cos h(b*c*x*log(F) - d)**3/8 + 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*lo g(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 + 3*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 - 3*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)/...
Time = 0.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 3 \, e x + 3 \, d\right )}}{8 \, {\left (b c \log \left (F\right ) + 3 \, e\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{8 \, {\left (b c \log \left (F\right ) + e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{8 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 3 \, e x\right )}}{8 \, {\left (b c e^{\left (3 \, d\right )} \log \left (F\right ) - 3 \, e e^{\left (3 \, d\right )}\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^2*sinh(e*x+d),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))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1211, normalized size of antiderivative = 9.17 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^2*sinh(e*x+d),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*l og(abs(F)) + (b*c*log(abs(F)) + e)*x + d) - 1/4*(2*(b*c*log(abs(F)) - e...
Time = 3.41 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.23 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (3\,e^3\,{\mathrm {cosh}\left (d+e\,x\right )}^3-2\,b\,c\,e^2\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )-3\,b^2\,c^2\,e\,{\mathrm {cosh}\left (d+e\,x\right )}^3\,{\ln \left (F\right )}^2+b^3\,c^3\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \left (F\right )}^3+2\,b^2\,c^2\,e\,\mathrm {cosh}\left (d+e\,x\right )\,{\ln \left (F\right )}^2-b\,c\,e^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4} \] Input:
int(F^(c*(a + b*x))*cosh(d + e*x)^2*sinh(d + e*x),x)
Output:
(F^(a*c + b*c*x)*(3*e^3*cosh(d + e*x)^3 - 2*b*c*e^2*sinh(d + e*x)*log(F) - 3*b^2*c^2*e*cosh(d + e*x)^3*log(F)^2 + b^3*c^3*cosh(d + e*x)^2*sinh(d + e *x)*log(F)^3 + 2*b^2*c^2*e*cosh(d + e*x)*log(F)^2 - b*c*e^2*cosh(d + e*x)^ 2*sinh(d + e*x)*log(F)))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e^2*log(F) ^2)
\[ \int F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cosh \left (e x +d \right )^{2} \sinh \left (e x +d \right )d x \right ) \] Input:
int(F^(c*(b*x+a))*cosh(e*x+d)^2*sinh(e*x+d),x)
Output:
f**(a*c)*int(f**(b*c*x)*cosh(d + e*x)**2*sinh(d + e*x),x)