\(\int F^{c (a+b x)} \text {csch}(d+e x) \text {sech}(d+e x) \, dx\) [120]

Optimal result

Integrand size = 22, antiderivative size = 72 \[ \int F^{c (a+b x)} \text {csch}(d+e x) \text {sech}(d+e x) \, dx=-\frac {4 e^{2 d+2 e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} \left (2+\frac {b c \log (F)}{e}\right ),\frac {1}{4} \left (6+\frac {b c \log (F)}{e}\right ),e^{4 d+4 e x}\right )}{2 e+b c \log (F)} \] Output:

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

Mathematica [A] (verified)

Time = 2.98 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33 \[ \int F^{c (a+b x)} \text {csch}(d+e x) \text {sech}(d+e x) \, dx=\frac {2 F^{c (a+b x)} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{2 e},1+\frac {b c \log (F)}{2 e},-e^{2 (d+e x)}\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{2 e},1+\frac {b c \log (F)}{2 e},e^{2 (d+e x)}\right )\right )}{b c \log (F)} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(72)=144\).

Time = 0.63 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 6037

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*F^(a*c + b*c*x)*Hypergeometric2F1[1, -1/2*(b*c*Log[F])/e, 1 - (b*c*Log 
[F])/(2*e), -E^(-2*(d + e*x))])/(b*c*Log[F]) - (E^(2*d + 2*e*x)*F^(a*c + b 
*c*x)*Hypergeometric2F1[1, 2 + (b*c*Log[F])/e, 3 + (b*c*Log[F])/e, -E^(d + 
 e*x)])/(2*e + b*c*Log[F]) - (E^(2*d + 2*e*x)*F^(a*c + b*c*x)*Hypergeometr 
ic2F1[1, 2 + (b*c*Log[F])/e, 3 + (b*c*Log[F])/e, E^(d + e*x)])/(2*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

16*F^(a*c)*e*integrate(e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + (b*c*e 
^(8*d)*log(F) - 2*e*e^(8*d))*e^(8*e*x) - 2*(b*c*e^(4*d)*log(F) - 2*e*e^(4* 
d))*e^(4*e*x) - 2*e), x) - 4*F^(a*c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*l 
og(F) - (b*c*e^(4*d)*log(F) - 2*e*e^(4*d))*e^(4*e*x) - 2*e)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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