\(\int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx\) [136]

Optimal result

Integrand size = 26, antiderivative size = 72 \[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=-\frac {64 e^{6 d+6 e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{4} \left (6+\frac {b c \log (F)}{e}\right ),\frac {1}{4} \left (10+\frac {b c \log (F)}{e}\right ),e^{4 d+4 e x}\right )}{6 e+b c \log (F)} \] Output:

-64*exp(6*e*x+6*d)*F^(c*(b*x+a))*hypergeom([3, 3/2+1/4*b*c*ln(F)/e],[5/2+1 
/4*b*c*ln(F)/e],exp(4*e*x+4*d))/(6*e+b*c*ln(F))
 

Mathematica [A] (verified)

Time = 2.45 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.00 \[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} \left (8 e^{\frac {(d+e x) (2 e+b c \log (F))}{e}} \coth (d+e x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} \left (2+\frac {b c \log (F)}{e}\right ),\frac {1}{4} \left (6+\frac {b c \log (F)}{e}\right ),e^{4 (d+e x)}\right ) (2 e-b c \log (F))-2 F^{\frac {b c (d+e x)}{e}} \text {csch}^2(d+e x) (2 e \coth (2 (d+e x))+b c \log (F))\right ) \tanh (d+e x)}{4 e^2} \] Input:

Integrate[F^(c*(a + b*x))*Csch[d + e*x]^3*Sech[d + e*x]^3,x]
 

Output:

(F^(c*(a - (b*d)/e))*(8*E^(((d + e*x)*(2*e + b*c*Log[F]))/e)*Coth[d + e*x] 
*Hypergeometric2F1[1, (2 + (b*c*Log[F])/e)/4, (6 + (b*c*Log[F])/e)/4, E^(4 
*(d + e*x))]*(2*e - b*c*Log[F]) - 2*F^((b*c*(d + e*x))/e)*Csch[d + e*x]^2* 
(2*e*Coth[2*(d + e*x)] + b*c*Log[F]))*Tanh[d + e*x])/(4*e^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(594\) vs. \(2(72)=144\).

Time = 1.62 (sec) , antiderivative size = 594, normalized size of antiderivative = 8.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6037, 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.

\(\displaystyle \int \text {csch}^3(d+e x) \text {sech}^3(d+e x) F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 6037

\(\displaystyle \int \left (\frac {21 e^{6 d+6 e x} F^{a c+b c x}}{2 \left (e^{d+e x}-1\right )}-\frac {21 e^{6 d+6 e x} F^{a c+b c x}}{2 \left (e^{d+e x}+1\right )}-\frac {12 e^{6 d+6 e x} F^{a c+b c x}}{e^{2 d+2 e x}+1}-\frac {9 e^{6 d+6 e x} F^{a c+b c x}}{2 \left (e^{d+e x}-1\right )^2}-\frac {9 e^{6 d+6 e x} F^{a c+b c x}}{2 \left (e^{d+e x}+1\right )^2}-\frac {12 e^{6 d+6 e x} F^{a c+b c x}}{\left (e^{2 d+2 e x}+1\right )^2}+\frac {e^{6 d+6 e x} F^{a c+b c x}}{\left (e^{d+e x}-1\right )^3}-\frac {e^{6 d+6 e x} F^{a c+b c x}}{\left (e^{d+e x}+1\right )^3}-\frac {8 e^{6 d+6 e x} F^{a c+b c x}}{\left (e^{2 d+2 e x}+1\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8 F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,-\frac {b c \log (F)}{2 e},1-\frac {b c \log (F)}{2 e},-e^{-2 (d+e x)}\right )}{b c \log (F)}-\frac {12 e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (\frac {b c \log (F)}{e}+6\right ),\frac {1}{2} \left (\frac {b c \log (F)}{e}+8\right ),-e^{2 (d+e x)}\right )}{b c \log (F)+6 e}-\frac {21 e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e}+6,\frac {b c \log (F)}{e}+7,-e^{d+e x}\right )}{2 (b c \log (F)+6 e)}-\frac {21 e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e}+6,\frac {b c \log (F)}{e}+7,e^{d+e x}\right )}{2 (b c \log (F)+6 e)}-\frac {12 e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (\frac {b c \log (F)}{e}+6\right ),\frac {1}{2} \left (\frac {b c \log (F)}{e}+8\right ),-e^{2 (d+e x)}\right )}{b c \log (F)+6 e}-\frac {9 e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {b c \log (F)}{e}+6,\frac {b c \log (F)}{e}+7,-e^{d+e x}\right )}{2 (b c \log (F)+6 e)}-\frac {9 e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {b c \log (F)}{e}+6,\frac {b c \log (F)}{e}+7,e^{d+e x}\right )}{2 (b c \log (F)+6 e)}-\frac {e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,\frac {b c \log (F)}{e}+6,\frac {b c \log (F)}{e}+7,-e^{d+e x}\right )}{b c \log (F)+6 e}-\frac {e^{6 d+6 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (3,\frac {b c \log (F)}{e}+6,\frac {b c \log (F)}{e}+7,e^{d+e x}\right )}{b c \log (F)+6 e}\)

Input:

Int[F^(c*(a + b*x))*Csch[d + e*x]^3*Sech[d + e*x]^3,x]
 

Output:

(-8*F^(a*c + b*c*x)*Hypergeometric2F1[3, -1/2*(b*c*Log[F])/e, 1 - (b*c*Log 
[F])/(2*e), -E^(-2*(d + e*x))])/(b*c*Log[F]) - (12*E^(6*d + 6*e*x)*F^(a*c 
+ b*c*x)*Hypergeometric2F1[1, (6 + (b*c*Log[F])/e)/2, (8 + (b*c*Log[F])/e) 
/2, -E^(2*(d + e*x))])/(6*e + b*c*Log[F]) - (21*E^(6*d + 6*e*x)*F^(a*c + b 
*c*x)*Hypergeometric2F1[1, 6 + (b*c*Log[F])/e, 7 + (b*c*Log[F])/e, -E^(d + 
 e*x)])/(2*(6*e + b*c*Log[F])) - (21*E^(6*d + 6*e*x)*F^(a*c + b*c*x)*Hyper 
geometric2F1[1, 6 + (b*c*Log[F])/e, 7 + (b*c*Log[F])/e, E^(d + e*x)])/(2*( 
6*e + b*c*Log[F])) - (12*E^(6*d + 6*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1 
[2, (6 + (b*c*Log[F])/e)/2, (8 + (b*c*Log[F])/e)/2, -E^(2*(d + e*x))])/(6* 
e + b*c*Log[F]) - (9*E^(6*d + 6*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2, 
6 + (b*c*Log[F])/e, 7 + (b*c*Log[F])/e, -E^(d + e*x)])/(2*(6*e + b*c*Log[F 
])) - (9*E^(6*d + 6*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2, 6 + (b*c*Log 
[F])/e, 7 + (b*c*Log[F])/e, E^(d + e*x)])/(2*(6*e + b*c*Log[F])) - (E^(6*d 
 + 6*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[3, 6 + (b*c*Log[F])/e, 7 + (b* 
c*Log[F])/e, -E^(d + e*x)])/(6*e + b*c*Log[F]) - (E^(6*d + 6*e*x)*F^(a*c + 
 b*c*x)*Hypergeometric2F1[3, 6 + (b*c*Log[F])/e, 7 + (b*c*Log[F])/e, E^(d 
+ e*x)])/(6*e + b*c*Log[F])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6037

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[( 
d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^(c*(a + b*x)), 
 G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[ 
m, 0] && IGtQ[n, 0] && HyperbolicQ[G] && HyperbolicQ[H]
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \operatorname {csch}\left (e x +d \right )^{3} \operatorname {sech}\left (e x +d \right )^{3}d x\]

Input:

int(F^(c*(b*x+a))*csch(e*x+d)^3*sech(e*x+d)^3,x)
 

Output:

int(F^(c*(b*x+a))*csch(e*x+d)^3*sech(e*x+d)^3,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{3} \operatorname {sech}\left (e x + d\right )^{3} \,d x } \] Input:

integrate(F^(c*(b*x+a))*csch(e*x+d)^3*sech(e*x+d)^3,x, algorithm="fricas")
 

Output:

integral(F^(b*c*x + a*c)*csch(e*x + d)^3*sech(e*x + d)^3, x)
 

Sympy [F]

\[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=\int F^{c \left (a + b x\right )} \operatorname {csch}^{3}{\left (d + e x \right )} \operatorname {sech}^{3}{\left (d + e x \right )}\, dx \] Input:

integrate(F**(c*(b*x+a))*csch(e*x+d)**3*sech(e*x+d)**3,x)
 

Output:

Integral(F**(c*(a + b*x))*csch(d + e*x)**3*sech(d + e*x)**3, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{3} \operatorname {sech}\left (e x + d\right )^{3} \,d x } \] Input:

integrate(F^(c*(b*x+a))*csch(e*x+d)^3*sech(e*x+d)^3,x, algorithm="maxima")
 

Output:

768*(F^(a*c)*b*c*e*e^(2*d)*log(F) + 2*F^(a*c)*e^2*e^(2*d))*integrate(e^(b* 
c*x*log(F) + 2*e*x)/(b^2*c^2*log(F)^2 - 16*b*c*e*log(F) + 60*e^2 + (b^2*c^ 
2*e^(16*d)*log(F)^2 - 16*b*c*e*e^(16*d)*log(F) + 60*e^2*e^(16*d))*e^(16*e* 
x) - 4*(b^2*c^2*e^(12*d)*log(F)^2 - 16*b*c*e*e^(12*d)*log(F) + 60*e^2*e^(1 
2*d))*e^(12*e*x) + 6*(b^2*c^2*e^(8*d)*log(F)^2 - 16*b*c*e*e^(8*d)*log(F) + 
 60*e^2*e^(8*d))*e^(8*e*x) - 4*(b^2*c^2*e^(4*d)*log(F)^2 - 16*b*c*e*e^(4*d 
)*log(F) + 60*e^2*e^(4*d))*e^(4*e*x)), x) - 64*(12*F^(a*c)*e*e^(2*e*x + 2* 
d) + (F^(a*c)*b*c*e^(6*d)*log(F) - 10*F^(a*c)*e*e^(6*d))*e^(6*e*x))*F^(b*c 
*x)/(b^2*c^2*log(F)^2 - 16*b*c*e*log(F) + 60*e^2 - (b^2*c^2*e^(12*d)*log(F 
)^2 - 16*b*c*e*e^(12*d)*log(F) + 60*e^2*e^(12*d))*e^(12*e*x) + 3*(b^2*c^2* 
e^(8*d)*log(F)^2 - 16*b*c*e*e^(8*d)*log(F) + 60*e^2*e^(8*d))*e^(8*e*x) - 3 
*(b^2*c^2*e^(4*d)*log(F)^2 - 16*b*c*e*e^(4*d)*log(F) + 60*e^2*e^(4*d))*e^( 
4*e*x))
 

Giac [F]

\[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{3} \operatorname {sech}\left (e x + d\right )^{3} \,d x } \] Input:

integrate(F^(c*(b*x+a))*csch(e*x+d)^3*sech(e*x+d)^3,x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)*csch(e*x + d)^3*sech(e*x + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\mathrm {cosh}\left (d+e\,x\right )}^3\,{\mathrm {sinh}\left (d+e\,x\right )}^3} \,d x \] Input:

int(F^(c*(a + b*x))/(cosh(d + e*x)^3*sinh(d + e*x)^3),x)
 

Output:

int(F^(c*(a + b*x))/(cosh(d + e*x)^3*sinh(d + e*x)^3), x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \text {csch}^3(d+e x) \text {sech}^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \mathrm {csch}\left (e x +d \right )^{3} \mathrm {sech}\left (e x +d \right )^{3}d x \right ) \] Input:

int(F^(c*(b*x+a))*csch(e*x+d)^3*sech(e*x+d)^3,x)
 

Output:

f**(a*c)*int(f**(b*c*x)*csch(d + e*x)**3*sech(d + e*x)**3,x)