Integrand size = 26, antiderivative size = 139 \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=-\frac {3 e^{-2 d-2 e x} F^{c (a+b x)}}{64 (2 e-b c \log (F))}+\frac {e^{-6 d-6 e x} F^{c (a+b x)}}{64 (6 e-b c \log (F))}-\frac {3 e^{2 d+2 e x} F^{c (a+b x)}}{64 (2 e+b c \log (F))}+\frac {e^{6 d+6 e x} F^{c (a+b x)}}{64 (6 e+b c \log (F))} \] Output:
-3*exp(-2*e*x-2*d)*F^(c*(b*x+a))/(128*e-64*b*c*ln(F))+exp(-6*e*x-6*d)*F^(c *(b*x+a))/(384*e-64*b*c*ln(F))-3*exp(2*e*x+2*d)*F^(c*(b*x+a))/(128*e+64*b* c*ln(F))+exp(6*e*x+6*d)*F^(c*(b*x+a))/(384*e+64*b*c*ln(F))
Time = 1.01 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.17 \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (6 \cosh (6 (d+e x)) \left (4 e^3-b^2 c^2 e \log ^2(F)\right )+6 \cosh (2 (d+e x)) \left (-36 e^3+b^2 c^2 e \log ^2(F)\right )+2 b c \log (F) \left (52 e^2-b^2 c^2 \log ^2(F)+\cosh (4 (d+e x)) \left (-4 e^2+b^2 c^2 \log ^2(F)\right )\right ) \sinh (2 (d+e x))\right )}{32 \left (144 e^4-40 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \] Input:
Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^3*Sinh[d + e*x]^3,x]
Output:
(F^(c*(a + b*x))*(6*Cosh[6*(d + e*x)]*(4*e^3 - b^2*c^2*e*Log[F]^2) + 6*Cos h[2*(d + e*x)]*(-36*e^3 + b^2*c^2*e*Log[F]^2) + 2*b*c*Log[F]*(52*e^2 - b^2 *c^2*Log[F]^2 + Cosh[4*(d + e*x)]*(-4*e^2 + b^2*c^2*Log[F]^2))*Sinh[2*(d + e*x)]))/(32*(144*e^4 - 40*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
Time = 0.36 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
\(\displaystyle \int \sinh ^3(d+e x) \cosh ^3(d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 6035 |
\(\displaystyle \int \left (\frac {1}{32} \sinh (6 d+6 e x) F^{c (a+b x)}-\frac {3}{32} \sinh (2 d+2 e x) F^{c (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 b c \log (F) \sinh (2 d+2 e x) F^{c (a+b x)}}{32 \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c \log (F) \sinh (6 d+6 e x) F^{c (a+b x)}}{32 \left (36 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {3 e \cosh (2 d+2 e x) F^{c (a+b x)}}{16 \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {3 e \cosh (6 d+6 e x) F^{c (a+b x)}}{16 \left (36 e^2-b^2 c^2 \log ^2(F)\right )}\) |
Input:
Int[F^(c*(a + b*x))*Cosh[d + e*x]^3*Sinh[d + e*x]^3,x]
Output:
(-3*e*F^(c*(a + b*x))*Cosh[2*d + 2*e*x])/(16*(4*e^2 - b^2*c^2*Log[F]^2)) + (3*e*F^(c*(a + b*x))*Cosh[6*d + 6*e*x])/(16*(36*e^2 - b^2*c^2*Log[F]^2)) + (3*b*c*F^(c*(a + b*x))*Log[F]*Sinh[2*d + 2*e*x])/(32*(4*e^2 - b^2*c^2*Lo g[F]^2)) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[6*d + 6*e*x])/(32*(36*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 = 0.57 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.37
\[\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{12 e x +12 d}-6 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{12 e x +12 d}-3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{8 e x +8 d}-4 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{12 e x +12 d}+6 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{8 e x +8 d}+24 e^{3} {\mathrm e}^{12 e x +12 d}+3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{4 e x +4 d}+108 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{8 e x +8 d}+6 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{4 e x +4 d}-216 e^{3} {\mathrm e}^{8 e x +8 d}-c^{3} b^{3} \ln \left (F \right )^{3}-108 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{4 e x +4 d}-6 c^{2} b^{2} \ln \left (F \right )^{2} e -216 e^{3} {\mathrm e}^{4 e x +4 d}+4 e^{2} b c \ln \left (F \right )+24 e^{3}\right ) {\mathrm e}^{-6 e x -6 d} F^{c \left (b x +a \right )}}{64 \left (b c \ln \left (F \right )-2 e \right ) \left (b c \ln \left (F \right )-6 e \right ) \left (2 e +b c \ln \left (F \right )\right ) \left (6 e +b c \ln \left (F \right )\right )}\]
Input:
int(F^(c*(b*x+a))*cosh(e*x+d)^3*sinh(e*x+d)^3,x)
Output:
1/64*(ln(F)^3*b^3*c^3*exp(12*e*x+12*d)-6*ln(F)^2*b^2*c^2*e*exp(12*e*x+12*d )-3*ln(F)^3*b^3*c^3*exp(8*e*x+8*d)-4*ln(F)*b*c*e^2*exp(12*e*x+12*d)+6*ln(F )^2*b^2*c^2*e*exp(8*e*x+8*d)+24*e^3*exp(12*e*x+12*d)+3*ln(F)^3*b^3*c^3*exp (4*e*x+4*d)+108*ln(F)*b*c*e^2*exp(8*e*x+8*d)+6*ln(F)^2*b^2*c^2*e*exp(4*e*x +4*d)-216*e^3*exp(8*e*x+8*d)-c^3*b^3*ln(F)^3-108*ln(F)*b*c*e^2*exp(4*e*x+4 *d)-6*c^2*b^2*ln(F)^2*e-216*e^3*exp(4*e*x+4*d)+4*e^2*b*c*ln(F)+24*e^3)/(b* c*ln(F)-2*e)*exp(-6*e*x-6*d)/(b*c*ln(F)-6*e)/(2*e+b*c*ln(F))/(6*e+b*c*ln(F ))*F^(c*(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 4202 vs. \(2 (125) = 250\).
Time = 0.20 (sec) , antiderivative size = 4202, normalized size of antiderivative = 30.23 \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^3*sinh(e*x+d)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=\text {Timed out} \] Input:
integrate(F**(c*(b*x+a))*cosh(e*x+d)**3*sinh(e*x+d)**3,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03 \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 6 \, e x + 6 \, d\right )}}{64 \, {\left (b c \log \left (F\right ) + 6 \, e\right )}} - \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{64 \, {\left (b c \log \left (F\right ) + 2 \, e\right )}} + \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{64 \, {\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 6 \, e x\right )}}{64 \, {\left (b c e^{\left (6 \, d\right )} \log \left (F\right ) - 6 \, e e^{\left (6 \, d\right )}\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^3*sinh(e*x+d)^3,x, algorithm="maxima")
Output:
1/64*F^(a*c)*e^(b*c*x*log(F) + 6*e*x + 6*d)/(b*c*log(F) + 6*e) - 3/64*F^(a *c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) + 3/64*F^(a*c)*e^(b* c*x*log(F) - 2*e*x)/(b*c*e^(2*d)*log(F) - 2*e*e^(2*d)) - 1/64*F^(a*c)*e^(b *c*x*log(F) - 6*e*x)/(b*c*e^(6*d)*log(F) - 6*e*e^(6*d))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1225, normalized size of antiderivative = 8.81 \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^3*sinh(e*x+d)^3,x, algorithm="giac")
Output:
1/32*(2*(b*c*log(abs(F)) + 6*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(a bs(F)) + 6*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)) + 6*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 6*e) *x + 6*d) + 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)/(64*I*pi*b*c*sgn(F) - 64*I*pi*b*c + 128*b*c*log(abs (F)) + 768*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)/(-64*I*pi*b*c*sgn(F) + 64*I*pi*b*c + 128*b*c*log( abs(F)) + 768*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 6*e)*x + 6*d) - 3/32*(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(a bs(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) + 3*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)/(64*I*pi*b*c*sgn(F) - 64*I*pi*b*c + 128*b*c*log( abs(F)) + 256*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)/(-64*I*pi*b*c*sgn(F) + 64*I*pi*b*c + 128*b*c*l og(abs(F)) + 256*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2...
Time = 4.06 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.77 \[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=-\frac {e^3\,\left (\frac {27\,F^{a\,c+b\,c\,x}\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )}{4}-\frac {3\,F^{a\,c+b\,c\,x}\,\mathrm {cosh}\left (6\,d+6\,e\,x\right )}{4}\right )+b^3\,c^3\,{\ln \left (F\right )}^3\,\left (\frac {3\,F^{a\,c+b\,c\,x}\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )}{32}-\frac {F^{a\,c+b\,c\,x}\,\mathrm {sinh}\left (6\,d+6\,e\,x\right )}{32}\right )-b^2\,c^2\,e\,{\ln \left (F\right )}^2\,\left (\frac {3\,F^{a\,c+b\,c\,x}\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )}{16}-\frac {3\,F^{a\,c+b\,c\,x}\,\mathrm {cosh}\left (6\,d+6\,e\,x\right )}{16}\right )-b\,c\,e^2\,\ln \left (F\right )\,\left (\frac {27\,F^{a\,c+b\,c\,x}\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )}{8}-\frac {F^{a\,c+b\,c\,x}\,\mathrm {sinh}\left (6\,d+6\,e\,x\right )}{8}\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-40\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+144\,e^4} \] Input:
int(F^(c*(a + b*x))*cosh(d + e*x)^3*sinh(d + e*x)^3,x)
Output:
-(e^3*((27*F^(a*c + b*c*x)*cosh(2*d + 2*e*x))/4 - (3*F^(a*c + b*c*x)*cosh( 6*d + 6*e*x))/4) + b^3*c^3*log(F)^3*((3*F^(a*c + b*c*x)*sinh(2*d + 2*e*x)) /32 - (F^(a*c + b*c*x)*sinh(6*d + 6*e*x))/32) - b^2*c^2*e*log(F)^2*((3*F^( a*c + b*c*x)*cosh(2*d + 2*e*x))/16 - (3*F^(a*c + b*c*x)*cosh(6*d + 6*e*x)) /16) - b*c*e^2*log(F)*((27*F^(a*c + b*c*x)*sinh(2*d + 2*e*x))/8 - (F^(a*c + b*c*x)*sinh(6*d + 6*e*x))/8))/(144*e^4 + b^4*c^4*log(F)^4 - 40*b^2*c^2*e ^2*log(F)^2)
\[ \int F^{c (a+b x)} \cosh ^3(d+e x) \sinh ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cosh \left (e x +d \right )^{3} \sinh \left (e x +d \right )^{3}d x \right ) \] Input:
int(F^(c*(b*x+a))*cosh(e*x+d)^3*sinh(e*x+d)^3,x)
Output:
f**(a*c)*int(f**(b*c*x)*cosh(d + e*x)**3*sinh(d + e*x)**3,x)