Integrand size = 24, antiderivative size = 139 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=-\frac {e^{-2 d-2 e x} F^{c (a+b x)}}{8 (2 e-b c \log (F))}+\frac {e^{-4 d-4 e x} F^{c (a+b x)}}{16 (4 e-b c \log (F))}-\frac {e^{2 d+2 e x} F^{c (a+b x)}}{8 (2 e+b c \log (F))}+\frac {e^{4 d+4 e x} F^{c (a+b x)}}{16 (4 e+b c \log (F))} \] Output:
-1/8*exp(-2*e*x-2*d)*F^(c*(b*x+a))/(2*e-b*c*ln(F))+exp(-4*e*x-4*d)*F^(c*(b *x+a))/(64*e-16*b*c*ln(F))-exp(2*e*x+2*d)*F^(c*(b*x+a))/(16*e+8*b*c*ln(F)) +exp(4*e*x+4*d)*F^(c*(b*x+a))/(64*e+16*b*c*ln(F))
Time = 1.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=\frac {1}{8} F^{c (a+b x)} \left (\frac {-4 e \cosh (2 (d+e x))+2 b c \log (F) \sinh (2 (d+e x))}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {4 e \cosh (4 (d+e x))-b c \log (F) \sinh (4 (d+e x))}{16 e^2-b^2 c^2 \log ^2(F)}\right ) \] Input:
Integrate[F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^3,x]
Output:
(F^(c*(a + b*x))*((-4*e*Cosh[2*(d + e*x)] + 2*b*c*Log[F]*Sinh[2*(d + e*x)] )/(4*e^2 - b^2*c^2*Log[F]^2) + (4*e*Cosh[4*(d + e*x)] - b*c*Log[F]*Sinh[4* (d + e*x)])/(16*e^2 - b^2*c^2*Log[F]^2)))/8
Time = 0.45 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29, 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.
\(\displaystyle \int \sinh ^3(d+e x) \cosh (d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 6035 |
\(\displaystyle \int \left (\frac {1}{8} \sinh (4 d+4 e x) F^{c (a+b x)}-\frac {1}{4} \sinh (2 d+2 e x) F^{c (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b c \log (F) \sinh (2 d+2 e x) F^{c (a+b x)}}{4 \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c \log (F) \sinh (4 d+4 e x) F^{c (a+b x)}}{8 \left (16 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {e \cosh (2 d+2 e x) F^{c (a+b x)}}{2 \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {e \cosh (4 d+4 e x) F^{c (a+b x)}}{2 \left (16 e^2-b^2 c^2 \log ^2(F)\right )}\) |
Input:
Int[F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^3,x]
Output:
-1/2*(e*F^(c*(a + b*x))*Cosh[2*d + 2*e*x])/(4*e^2 - b^2*c^2*Log[F]^2) + (e *F^(c*(a + b*x))*Cosh[4*d + 4*e*x])/(2*(16*e^2 - b^2*c^2*Log[F]^2)) + (b*c *F^(c*(a + b*x))*Log[F]*Sinh[2*d + 2*e*x])/(4*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[4*d + 4*e*x])/(8*(16*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. \(328\) vs. \(2(127)=254\).
Time = 0.66 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.37
\[\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{8 e x +8 d}-2 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{6 e x +6 d}-4 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{8 e x +8 d}+4 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{6 e x +6 d}-4 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{8 e x +8 d}+2 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{2 e x +2 d}+32 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{6 e x +6 d}+16 e^{3} {\mathrm e}^{8 e x +8 d}-c^{3} b^{3} \ln \left (F \right )^{3}+4 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{2 e x +2 d}-64 e^{3} {\mathrm e}^{6 e x +6 d}-4 c^{2} b^{2} \ln \left (F \right )^{2} e -32 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{2 e x +2 d}+4 e^{2} b c \ln \left (F \right )-64 e^{3} {\mathrm e}^{2 e x +2 d}+16 e^{3}\right ) {\mathrm e}^{-4 e x -4 d} F^{c \left (b x +a \right )}}{16 \left (b c \ln \left (F \right )-2 e \right ) \left (b c \ln \left (F \right )-4 e \right ) \left (2 e +b c \ln \left (F \right )\right ) \left (b c \ln \left (F \right )+4 e \right )}\]
Input:
int(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)^3,x)
Output:
1/16*(ln(F)^3*b^3*c^3*exp(8*e*x+8*d)-2*ln(F)^3*b^3*c^3*exp(6*e*x+6*d)-4*ln (F)^2*b^2*c^2*e*exp(8*e*x+8*d)+4*ln(F)^2*b^2*c^2*e*exp(6*e*x+6*d)-4*ln(F)* b*c*e^2*exp(8*e*x+8*d)+2*ln(F)^3*b^3*c^3*exp(2*e*x+2*d)+32*ln(F)*b*c*e^2*e xp(6*e*x+6*d)+16*e^3*exp(8*e*x+8*d)-c^3*b^3*ln(F)^3+4*ln(F)^2*b^2*c^2*e*ex p(2*e*x+2*d)-64*e^3*exp(6*e*x+6*d)-4*c^2*b^2*ln(F)^2*e-32*ln(F)*b*c*e^2*ex p(2*e*x+2*d)+4*e^2*b*c*ln(F)-64*e^3*exp(2*e*x+2*d)+16*e^3)/(b*c*ln(F)-2*e) *exp(-4*e*x-4*d)/(b*c*ln(F)-4*e)/(2*e+b*c*ln(F))/(b*c*ln(F)+4*e)*F^(c*(b*x +a))
Leaf count of result is larger than twice the leaf count of optimal. 2900 vs. \(2 (125) = 250\).
Time = 0.20 (sec) , antiderivative size = 2900, normalized size of antiderivative = 20.86 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)^3,x, algorithm="fricas")
Output:
1/16*((16*e^3*cosh(e*x + d)^8 - 64*e^3*cosh(e*x + d)^6 + (b^3*c^3*log(F)^3 - 4*b^2*c^2*e*log(F)^2 - 4*b*c*e^2*log(F) + 16*e^3)*sinh(e*x + d)^8 + 8*( b^3*c^3*cosh(e*x + d)*log(F)^3 - 4*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - 4*b* c*e^2*cosh(e*x + d)*log(F) + 16*e^3*cosh(e*x + d))*sinh(e*x + d)^7 + 2*(22 4*e^3*cosh(e*x + d)^2 + (14*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*log(F)^3 - 32*e^3 - 2*(28*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - 8*(7*b*c* e^2*cosh(e*x + d)^2 - 2*b*c*e^2)*log(F))*sinh(e*x + d)^6 + 4*(224*e^3*cosh (e*x + d)^3 - 96*e^3*cosh(e*x + d) + (14*b^3*c^3*cosh(e*x + d)^3 - 3*b^3*c ^3*cosh(e*x + d))*log(F)^3 - 2*(28*b^2*c^2*e*cosh(e*x + d)^3 - 3*b^2*c^2*e *cosh(e*x + d))*log(F)^2 - 8*(7*b*c*e^2*cosh(e*x + d)^3 - 6*b*c*e^2*cosh(e *x + d))*log(F))*sinh(e*x + d)^5 - 64*e^3*cosh(e*x + d)^2 + 10*(112*e^3*co sh(e*x + d)^4 - 96*e^3*cosh(e*x + d)^2 + (7*b^3*c^3*cosh(e*x + d)^4 - 3*b^ 3*c^3*cosh(e*x + d)^2)*log(F)^3 - 2*(14*b^2*c^2*e*cosh(e*x + d)^4 - 3*b^2* c^2*e*cosh(e*x + d)^2)*log(F)^2 - 4*(7*b*c*e^2*cosh(e*x + d)^4 - 12*b*c*e^ 2*cosh(e*x + d)^2)*log(F))*sinh(e*x + d)^4 + (b^3*c^3*cosh(e*x + d)^8 - 2* b^3*c^3*cosh(e*x + d)^6 + 2*b^3*c^3*cosh(e*x + d)^2 - b^3*c^3)*log(F)^3 + 8*(112*e^3*cosh(e*x + d)^5 - 160*e^3*cosh(e*x + d)^3 + (7*b^3*c^3*cosh(e*x + d)^5 - 5*b^3*c^3*cosh(e*x + d)^3)*log(F)^3 - 2*(14*b^2*c^2*e*cosh(e*x + d)^5 - 5*b^2*c^2*e*cosh(e*x + d)^3)*log(F)^2 - 4*(7*b*c*e^2*cosh(e*x + d) ^5 - 20*b*c*e^2*cosh(e*x + d)^3)*log(F))*sinh(e*x + d)^3 + 16*e^3 - 4*(...
Leaf count of result is larger than twice the leaf count of optimal. 1926 vs. \(2 (124) = 248\).
Time = 24.10 (sec) , antiderivative size = 1926, normalized size of antiderivative = 13.86 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)**3,x)
Output:
Piecewise((x*sinh(d)**3*cosh(d), Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sinh(d) **3*cosh(d), Eq(b, 0) & Eq(e, 0)), (x*sinh(d)**3*cosh(d), Eq(c, 0) & Eq(e, 0)), (F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/2 - d)**4/8 - F**(a*c + b*c*x) *x*sinh(b*c*x*log(F)/2 - d)**3*cosh(b*c*x*log(F)/2 - d)/4 + F**(a*c + b*c* x)*x*sinh(b*c*x*log(F)/2 - d)*cosh(b*c*x*log(F)/2 - d)**3/4 - F**(a*c + b* c*x)*x*cosh(b*c*x*log(F)/2 - d)**4/8 - 7*F**(a*c + b*c*x)*sinh(b*c*x*log(F )/2 - d)**4/(24*b*c*log(F)) + F**(a*c + b*c*x)*sinh(b*c*x*log(F)/2 - d)**3 *cosh(b*c*x*log(F)/2 - d)/(3*b*c*log(F)) - F**(a*c + b*c*x)*sinh(b*c*x*log (F)/2 - d)**2*cosh(b*c*x*log(F)/2 - d)**2/(2*b*c*log(F)) + F**(a*c + b*c*x )*cosh(b*c*x*log(F)/2 - d)**4/(8*b*c*log(F)), Eq(e, -b*c*log(F)/2)), (F**( a*c + b*c*x)*x*sinh(b*c*x*log(F)/4 - d)**4/16 - F**(a*c + b*c*x)*x*sinh(b* c*x*log(F)/4 - d)**3*cosh(b*c*x*log(F)/4 - d)/4 + 3*F**(a*c + b*c*x)*x*sin h(b*c*x*log(F)/4 - d)**2*cosh(b*c*x*log(F)/4 - d)**2/8 - F**(a*c + b*c*x)* x*sinh(b*c*x*log(F)/4 - d)*cosh(b*c*x*log(F)/4 - d)**3/4 + F**(a*c + b*c*x )*x*cosh(b*c*x*log(F)/4 - d)**4/16 + F**(a*c + b*c*x)*sinh(b*c*x*log(F)/4 - d)**4/(6*b*c*log(F)) - 11*F**(a*c + b*c*x)*sinh(b*c*x*log(F)/4 - d)**3*c osh(b*c*x*log(F)/4 - d)/(12*b*c*log(F)) + 5*F**(a*c + b*c*x)*sinh(b*c*x*lo g(F)/4 - d)*cosh(b*c*x*log(F)/4 - d)**3/(12*b*c*log(F)) - F**(a*c + b*c*x) *cosh(b*c*x*log(F)/4 - d)**4/(6*b*c*log(F)), Eq(e, -b*c*log(F)/4)), (-F**( a*c + b*c*x)*x*sinh(b*c*x*log(F)/4 + d)**4/16 + F**(a*c + b*c*x)*x*sinh...
Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 4 \, e x + 4 \, d\right )}}{16 \, {\left (b c \log \left (F\right ) + 4 \, e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{8 \, {\left (b c \log \left (F\right ) + 2 \, e\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{8 \, {\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 4 \, e x\right )}}{16 \, {\left (b c e^{\left (4 \, d\right )} \log \left (F\right ) - 4 \, e e^{\left (4 \, d\right )}\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)^3,x, algorithm="maxima")
Output:
1/16*F^(a*c)*e^(b*c*x*log(F) + 4*e*x + 4*d)/(b*c*log(F) + 4*e) - 1/8*F^(a* c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) + 1/8*F^(a*c)*e^(b*c* x*log(F) - 2*e*x)/(b*c*e^(2*d)*log(F) - 2*e*e^(2*d)) - 1/16*F^(a*c)*e^(b*c *x*log(F) - 4*e*x)/(b*c*e^(4*d)*log(F) - 4*e*e^(4*d))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1225, normalized size of antiderivative = 8.81 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)^3,x, algorithm="giac")
Output:
1/8*(2*(b*c*log(abs(F)) + 4*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)) + 4*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)) + 4*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 4*e)* x + 4*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs (F)) + 64*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs( F)) + 64*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 4*e)*x + 4*d) - 1/4*( 2*(b*c*log(abs(F)) + 2*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)) + 2*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)) + 2*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2 *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*s gn(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*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))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) + 1/4*(2*(...
Time = 4.97 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.42 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=-\frac {F^{a\,c+b\,c\,x}\,\left (8\,e^3\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )-2\,e^3\,\mathrm {cosh}\left (4\,d+4\,e\,x\right )+\frac {b^3\,c^3\,{\ln \left (F\right )}^3\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )}{4}-\frac {b^3\,c^3\,{\ln \left (F\right )}^3\,\mathrm {sinh}\left (4\,d+4\,e\,x\right )}{8}-\frac {b^2\,c^2\,e\,{\ln \left (F\right )}^2\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )}{2}+\frac {b^2\,c^2\,e\,{\ln \left (F\right )}^2\,\mathrm {cosh}\left (4\,d+4\,e\,x\right )}{2}-4\,b\,c\,e^2\,\ln \left (F\right )\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )+\frac {b\,c\,e^2\,\ln \left (F\right )\,\mathrm {sinh}\left (4\,d+4\,e\,x\right )}{2}\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-20\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+64\,e^4} \] Input:
int(F^(c*(a + b*x))*cosh(d + e*x)*sinh(d + e*x)^3,x)
Output:
-(F^(a*c + b*c*x)*(8*e^3*cosh(2*d + 2*e*x) - 2*e^3*cosh(4*d + 4*e*x) + (b^ 3*c^3*log(F)^3*sinh(2*d + 2*e*x))/4 - (b^3*c^3*log(F)^3*sinh(4*d + 4*e*x)) /8 - (b^2*c^2*e*log(F)^2*cosh(2*d + 2*e*x))/2 + (b^2*c^2*e*log(F)^2*cosh(4 *d + 4*e*x))/2 - 4*b*c*e^2*log(F)*sinh(2*d + 2*e*x) + (b*c*e^2*log(F)*sinh (4*d + 4*e*x))/2))/(64*e^4 + b^4*c^4*log(F)^4 - 20*b^2*c^2*e^2*log(F)^2)
\[ \int F^{c (a+b x)} \cosh (d+e x) \sinh ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cosh \left (e x +d \right ) \sinh \left (e x +d \right )^{3}d x \right ) \] Input:
int(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)^3,x)
Output:
f**(a*c)*int(f**(b*c*x)*cosh(d + e*x)*sinh(d + e*x)**3,x)