\(\int F^{c (a+b x)} (f \text {csch}(d-\frac {b c x \log (F)}{-2+n}))^n \, dx\) [119]

Optimal result

Integrand size = 29, antiderivative size = 130 \[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=-\frac {f^2 F^{c (a+b x)} (2-n) \left (f \text {csch}\left (d+\frac {b c x \log (F)}{2-n}\right )\right )^{-2+n}}{b c (1-n) \log (F)}+\frac {f F^{c (a+b x)} (2-n) \cosh \left (d+\frac {b c x \log (F)}{2-n}\right ) \left (f \text {csch}\left (d+\frac {b c x \log (F)}{2-n}\right )\right )^{-1+n}}{b c (1-n) \log (F)} \] Output:

-f^2*F^(c*(b*x+a))*(2-n)*(f*csch(d+b*c*x*ln(F)/(2-n)))^(-2+n)/b/c/(1-n)/ln 
(F)+f*F^(c*(b*x+a))*(2-n)*cosh(d+b*c*x*ln(F)/(2-n))*(f*csch(d+b*c*x*ln(F)/ 
(2-n)))^(-1+n)/b/c/(1-n)/ln(F)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.58 \[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=\frac {e^{-2 d} F^{c (a+b x)} \left (e^{2 d}-F^{\frac {2 b c x}{-2+n}}\right ) (-2+n) \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n}{2 b c (-1+n) \log (F)} \] Input:

Integrate[F^(c*(a + b*x))*(f*Csch[d - (b*c*x*Log[F])/(-2 + n)])^n,x]
 

Output:

(F^(c*(a + b*x))*(E^(2*d) - F^((2*b*c*x)/(-2 + n)))*(-2 + n)*(f*Csch[d - ( 
b*c*x*Log[F])/(-2 + n)])^n)/(2*b*c*E^(2*d)*(-1 + n)*Log[F])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {7271, 6012}

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.

\(\displaystyle \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{n-2}\right )\right )^n \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \text {csch}^{-n}\left (\frac {b c x \log (F)}{2-n}+d\right ) \left (f \text {csch}\left (\frac {b c x \log (F)}{2-n}+d\right )\right )^n \int F^{c (a+b x)} \text {csch}^n\left (d+\frac {b c x \log (F)}{2-n}\right )dx\)

\(\Big \downarrow \) 6012

\(\displaystyle \text {csch}^{-n}\left (\frac {b c x \log (F)}{2-n}+d\right ) \left (f \text {csch}\left (\frac {b c x \log (F)}{2-n}+d\right )\right )^n \left (\frac {(2-n) F^{c (a+b x)} \cosh \left (\frac {b c x \log (F)}{2-n}+d\right ) \text {csch}^{n-1}\left (\frac {b c x \log (F)}{2-n}+d\right )}{b c (1-n) \log (F)}-\frac {(2-n) F^{c (a+b x)} \text {csch}^{n-2}\left (\frac {b c x \log (F)}{2-n}+d\right )}{b c (1-n) \log (F)}\right )\)

Input:

Int[F^(c*(a + b*x))*(f*Csch[d - (b*c*x*Log[F])/(-2 + n)])^n,x]
 

Output:

((f*Csch[d + (b*c*x*Log[F])/(2 - n)])^n*(-((F^(c*(a + b*x))*(2 - n)*Csch[d 
 + (b*c*x*Log[F])/(2 - n)]^(-2 + n))/(b*c*(1 - n)*Log[F])) + (F^(c*(a + b* 
x))*(2 - n)*Cosh[d + (b*c*x*Log[F])/(2 - n)]*Csch[d + (b*c*x*Log[F])/(2 - 
n)]^(-1 + n))/(b*c*(1 - n)*Log[F])))/Csch[d + (b*c*x*Log[F])/(2 - n)]^n
 

Defintions of rubi rules used

rule 6012

Int[Csch[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symb 
ol] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Csch[d + e*x]^(n - 2)/(e^2*(n - 
1)*(n - 2))), x] - Simp[F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*(Cosh[d + e*x 
]/(e*(n - 1))), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 
- b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]
 

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (-f \,\operatorname {csch}\left (-d +\frac {b c x \ln \left (F \right )}{n -2}\right )\right )^{n}d x\]

Input:

int(F^(c*(b*x+a))*(-f*csch(-d+b*c*x*ln(F)/(n-2)))^n,x)
 

Output:

int(F^(c*(b*x+a))*(-f*csch(-d+b*c*x*ln(F)/(n-2)))^n,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (119) = 238\).

Time = 0.11 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.99 \[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=-\frac {{\left ({\left (n - 2\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) + {\left (n - 2\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )\right )} \cosh \left (n \log \left (-\frac {2 \, {\left (f \cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) + f \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )\right )}}{\cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )^{2} + 2 \, \cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) + \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )^{2} - 1}\right )\right ) + {\left ({\left (n - 2\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) + {\left (n - 2\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )\right )} \sinh \left (n \log \left (-\frac {2 \, {\left (f \cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) + f \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )\right )}}{\cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )^{2} + 2 \, \cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) + \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )^{2} - 1}\right )\right )}{{\left (b c n - b c\right )} \cosh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right ) \log \left (F\right ) - {\left (b c n - b c\right )} \log \left (F\right ) \sinh \left (\frac {b c x \log \left (F\right ) - d n + 2 \, d}{n - 2}\right )} \] Input:

integrate(F^(c*(b*x+a))*(-f*csch(-d+b*c*x*log(F)/(-2+n)))^n,x, algorithm=" 
fricas")
 

Output:

-(((n - 2)*cosh((b*c*x + a*c)*log(F))*sinh((b*c*x*log(F) - d*n + 2*d)/(n - 
 2)) + (n - 2)*sinh((b*c*x + a*c)*log(F))*sinh((b*c*x*log(F) - d*n + 2*d)/ 
(n - 2)))*cosh(n*log(-2*(f*cosh((b*c*x*log(F) - d*n + 2*d)/(n - 2)) + f*si 
nh((b*c*x*log(F) - d*n + 2*d)/(n - 2)))/(cosh((b*c*x*log(F) - d*n + 2*d)/( 
n - 2))^2 + 2*cosh((b*c*x*log(F) - d*n + 2*d)/(n - 2))*sinh((b*c*x*log(F) 
- d*n + 2*d)/(n - 2)) + sinh((b*c*x*log(F) - d*n + 2*d)/(n - 2))^2 - 1))) 
+ ((n - 2)*cosh((b*c*x + a*c)*log(F))*sinh((b*c*x*log(F) - d*n + 2*d)/(n - 
 2)) + (n - 2)*sinh((b*c*x + a*c)*log(F))*sinh((b*c*x*log(F) - d*n + 2*d)/ 
(n - 2)))*sinh(n*log(-2*(f*cosh((b*c*x*log(F) - d*n + 2*d)/(n - 2)) + f*si 
nh((b*c*x*log(F) - d*n + 2*d)/(n - 2)))/(cosh((b*c*x*log(F) - d*n + 2*d)/( 
n - 2))^2 + 2*cosh((b*c*x*log(F) - d*n + 2*d)/(n - 2))*sinh((b*c*x*log(F) 
- d*n + 2*d)/(n - 2)) + sinh((b*c*x*log(F) - d*n + 2*d)/(n - 2))^2 - 1)))) 
/((b*c*n - b*c)*cosh((b*c*x*log(F) - d*n + 2*d)/(n - 2))*log(F) - (b*c*n - 
 b*c)*log(F)*sinh((b*c*x*log(F) - d*n + 2*d)/(n - 2)))
 

Sympy [F]

\[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=\int F^{c \left (a + b x\right )} \left (- f \operatorname {csch}{\left (\frac {b c x \log {\left (F \right )}}{n - 2} - d \right )}\right )^{n}\, dx \] Input:

integrate(F**(c*(b*x+a))*(-f*csch(-d+b*c*x*ln(F)/(-2+n)))**n,x)
 

Output:

Integral(F**(c*(a + b*x))*(-f*csch(b*c*x*log(F)/(n - 2) - d))**n, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=\int { \left (-f \operatorname {csch}\left (\frac {b c x \log \left (F\right )}{n - 2} - d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(-f*csch(-d+b*c*x*log(F)/(-2+n)))^n,x, algorithm=" 
maxima")
 

Output:

integrate((-f*csch(b*c*x*log(F)/(n - 2) - d))^n*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=\int { \left (-f \operatorname {csch}\left (\frac {b c x \log \left (F\right )}{n - 2} - d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(-f*csch(-d+b*c*x*log(F)/(-2+n)))^n,x, algorithm=" 
giac")
 

Output:

integrate((-f*csch(b*c*x*log(F)/(n - 2) - d))^n*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

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

int(F^(c*(a + b*x))*(f/sinh(d - (b*c*x*log(F))/(n - 2)))^n,x)
 

Output:

int(F^(c*(a + b*x))*(f/sinh(d - (b*c*x*log(F))/(n - 2)))^n, x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \left (f \text {csch}\left (d-\frac {b c x \log (F)}{-2+n}\right )\right )^n \, dx=f^{a c +n} \left (-1\right )^{n} \left (\int f^{b c x} \mathrm {csch}\left (\frac {\mathrm {log}\left (f \right ) b c x -d n +2 d}{n -2}\right )^{n}d x \right ) \] Input:

int(F^(c*(b*x+a))*(-f*csch(-d+b*c*x*log(F)/(-2+n)))^n,x)
 

Output:

f**(a*c + n)*( - 1)**n*int(f**(b*c*x)*csch((log(f)*b*c*x - d*n + 2*d)/(n - 
 2))**n,x)