\(\int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx\) [128]

Optimal result

Integrand size = 26, antiderivative size = 72 \[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=\frac {16 e^{4 d+4 e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{4} \left (4+\frac {b c \log (F)}{e}\right ),\frac {1}{4} \left (8+\frac {b c \log (F)}{e}\right ),e^{4 d+4 e x}\right )}{4 e+b c \log (F)} \] Output:

16*exp(4*e*x+4*d)*F^(c*(b*x+a))*hypergeom([2, 1+1/4*b*c*ln(F)/e],[2+1/4*b* 
c*ln(F)/e],exp(4*e*x+4*d))/(b*c*ln(F)+4*e)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85 \[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=-\frac {2 F^{c (a+b x)} \left (-1+\coth (2 (d+e x))+2 \operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{4 e},1+\frac {b c \log (F)}{4 e},e^{4 (d+e x)}\right )\right )}{e} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(384\) vs. \(2(72)=144\).

Time = 1.00 (sec) , antiderivative size = 384, normalized size of antiderivative = 5.33, 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.

\(\displaystyle \int \text {csch}^2(d+e x) \text {sech}^2(d+e x) F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 6037

\(\displaystyle \int \left (-\frac {3 e^{4 d+4 e x} F^{a c+b c x}}{e^{d+e x}-1}+\frac {3 e^{4 d+4 e x} F^{a c+b c x}}{e^{d+e x}+1}+\frac {4 e^{4 d+4 e x} F^{a c+b c x}}{e^{2 d+2 e x}+1}+\frac {e^{4 d+4 e x} F^{a c+b c x}}{\left (e^{d+e x}-1\right )^2}+\frac {e^{4 d+4 e x} F^{a c+b c x}}{\left (e^{d+e x}+1\right )^2}+\frac {4 e^{4 d+4 e x} F^{a c+b c x}}{\left (e^{2 d+2 e x}+1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,-\frac {b c \log (F)}{2 e},1-\frac {b c \log (F)}{2 e},-e^{-2 (d+e x)}\right )}{b c \log (F)}+\frac {4 e^{4 d+4 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (\frac {b c \log (F)}{e}+4\right ),\frac {1}{2} \left (\frac {b c \log (F)}{e}+6\right ),-e^{2 (d+e x)}\right )}{b c \log (F)+4 e}+\frac {3 e^{4 d+4 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e}+4,\frac {b c \log (F)}{e}+5,-e^{d+e x}\right )}{b c \log (F)+4 e}+\frac {3 e^{4 d+4 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e}+4,\frac {b c \log (F)}{e}+5,e^{d+e x}\right )}{b c \log (F)+4 e}+\frac {e^{4 d+4 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {b c \log (F)}{e}+4,\frac {b c \log (F)}{e}+5,-e^{d+e x}\right )}{b c \log (F)+4 e}+\frac {e^{4 d+4 e x} F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (2,\frac {b c \log (F)}{e}+4,\frac {b c \log (F)}{e}+5,e^{d+e x}\right )}{b c \log (F)+4 e}\)

Input:

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

Output:

(4*F^(a*c + b*c*x)*Hypergeometric2F1[2, -1/2*(b*c*Log[F])/e, 1 - (b*c*Log[ 
F])/(2*e), -E^(-2*(d + e*x))])/(b*c*Log[F]) + (4*E^(4*d + 4*e*x)*F^(a*c + 
b*c*x)*Hypergeometric2F1[1, (4 + (b*c*Log[F])/e)/2, (6 + (b*c*Log[F])/e)/2 
, -E^(2*(d + e*x))])/(4*e + b*c*Log[F]) + (3*E^(4*d + 4*e*x)*F^(a*c + b*c* 
x)*Hypergeometric2F1[1, 4 + (b*c*Log[F])/e, 5 + (b*c*Log[F])/e, -E^(d + e* 
x)])/(4*e + b*c*Log[F]) + (3*E^(4*d + 4*e*x)*F^(a*c + b*c*x)*Hypergeometri 
c2F1[1, 4 + (b*c*Log[F])/e, 5 + (b*c*Log[F])/e, E^(d + e*x)])/(4*e + b*c*L 
og[F]) + (E^(4*d + 4*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2, 4 + (b*c*Lo 
g[F])/e, 5 + (b*c*Log[F])/e, -E^(d + e*x)])/(4*e + b*c*Log[F]) + (E^(4*d + 
 4*e*x)*F^(a*c + b*c*x)*Hypergeometric2F1[2, 4 + (b*c*Log[F])/e, 5 + (b*c* 
Log[F])/e, E^(d + e*x)])/(4*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]

\[\int F^{c \left (b x +a \right )} \operatorname {csch}\left (e x +d \right )^{2} \operatorname {sech}\left (e x +d \right )^{2}d x\]

Input:

int(F^(c*(b*x+a))*csch(e*x+d)^2*sech(e*x+d)^2,x)
 

Output:

int(F^(c*(b*x+a))*csch(e*x+d)^2*sech(e*x+d)^2,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{2} \operatorname {sech}\left (e x + d\right )^{2} \,d x } \] Input:

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

Output:

integral(F^(b*c*x + a*c)*csch(e*x + d)^2*sech(e*x + d)^2, x)
 

Sympy [F]

\[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=\int F^{c \left (a + b x\right )} \operatorname {csch}^{2}{\left (d + e x \right )} \operatorname {sech}^{2}{\left (d + e x \right )}\, dx \] Input:

integrate(F**(c*(b*x+a))*csch(e*x+d)**2*sech(e*x+d)**2,x)
 

Output:

Integral(F**(c*(a + b*x))*csch(d + e*x)**2*sech(d + e*x)**2, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{2} \operatorname {sech}\left (e x + d\right )^{2} \,d x } \] Input:

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

Output:

128*F^(a*c)*b*c*e*integrate(-F^(b*c*x)/(b^2*c^2*log(F)^2 - 12*b*c*e*log(F) 
 + 32*e^2 - (b^2*c^2*e^(12*d)*log(F)^2 - 12*b*c*e*e^(12*d)*log(F) + 32*e^2 
*e^(12*d))*e^(12*e*x) + 3*(b^2*c^2*e^(8*d)*log(F)^2 - 12*b*c*e*e^(8*d)*log 
(F) + 32*e^2*e^(8*d))*e^(8*e*x) - 3*(b^2*c^2*e^(4*d)*log(F)^2 - 12*b*c*e*e 
^(4*d)*log(F) + 32*e^2*e^(4*d))*e^(4*e*x)), x)*log(F) + 16*(8*F^(a*c)*e + 
(F^(a*c)*b*c*e^(4*d)*log(F) - 8*F^(a*c)*e*e^(4*d))*e^(4*e*x))*F^(b*c*x)/(b 
^2*c^2*log(F)^2 - 12*b*c*e*log(F) + 32*e^2 + (b^2*c^2*e^(8*d)*log(F)^2 - 1 
2*b*c*e*e^(8*d)*log(F) + 32*e^2*e^(8*d))*e^(8*e*x) - 2*(b^2*c^2*e^(4*d)*lo 
g(F)^2 - 12*b*c*e*e^(4*d)*log(F) + 32*e^2*e^(4*d))*e^(4*e*x))
 

Giac [F]

\[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{2} \operatorname {sech}\left (e x + d\right )^{2} \,d x } \] Input:

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

Output:

integrate(F^((b*x + a)*c)*csch(e*x + d)^2*sech(e*x + d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \text {sech}^2(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\mathrm {cosh}\left (d+e\,x\right )}^2\,{\mathrm {sinh}\left (d+e\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

int(F^(c*(b*x+a))*csch(e*x+d)^2*sech(e*x+d)^2,x)
 

Output:

f**(a*c)*int(f**(b*c*x)*csch(d + e*x)**2*sech(d + e*x)**2,x)