\(\int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx\) [138]

Optimal result

Integrand size = 34, antiderivative size = 44 \[ \int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\frac {F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)}}{b c \log (F)} \] Output:

F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2)/b/c/ln(F)
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\frac {F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)}}{b c \log (F)} \] Input:

Integrate[F^(c*(a + b*x))*Sqrt[g*Csch[d + e*x]]*Sqrt[f*Sinh[d + e*x]],x]
 

Output:

(F^(c*(a + b*x))*Sqrt[g*Csch[d + e*x]]*Sqrt[f*Sinh[d + e*x]])/(b*c*Log[F])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7271

\(\displaystyle \frac {\sqrt {g \text {csch}(d+e x)} \int F^{c (a+b x)} \sqrt {\text {csch}(d+e x)} \sqrt {f \sinh (d+e x)}dx}{\sqrt {\text {csch}(d+e x)}}\)

\(\Big \downarrow \) 7271

\(\displaystyle \sqrt {\sinh (d+e x)} \sqrt {\text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \sqrt {g \text {csch}(d+e x)} \int F^{c (a+b x)} \sqrt {\text {csch}(d+e x)} \sqrt {\sinh (d+e x)}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \sqrt {\sinh (d+e x)} \sqrt {\text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \sqrt {g \text {csch}(d+e x)} \int F^{a c+b x c} \sqrt {\text {csch}(d+e x)} \sqrt {\sinh (d+e x)}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \sqrt {\sinh (d+e x)} \sqrt {\text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \sqrt {g \text {csch}(d+e x)} \int F^{a c+b x c} \sqrt {\text {csch}(d+e x)} \sqrt {\sinh (d+e x)}dx\)

Input:

Int[F^(c*(a + b*x))*Sqrt[g*Csch[d + e*x]]*Sqrt[f*Sinh[d + e*x]],x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93

method	result	size
orering	\(\frac {F^{c \left (b x +a \right )} \sqrt {g \,\operatorname {csch}\left (e x +d \right )}\, \sqrt {f \sinh \left (e x +d \right )}}{b c \ln \left (F \right )}\)	\(41\)
risch	\(\frac {\sqrt {f \left (-1+{\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-e x -d}}\, \sqrt {\frac {g \,{\mathrm e}^{e x +d}}{-1+{\mathrm e}^{2 e x +2 d}}}\, F^{c \left (b x +a \right )}}{\ln \left (F \right ) b c}\)	\(68\)

Input:

int(F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2),x,method=_RE 
TURNVERBOSE)
 

Output:

F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2)/b/c/ln(F)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (40) = 80\).

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.34 \[ \int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\frac {\sqrt {2} \sqrt {f \sinh \left (e x + d\right )} \sqrt {\frac {g \cosh \left (e x + d\right ) + g \sinh \left (e x + d\right )}{\cosh \left (e x + d\right )^{2} + 2 \, \cosh \left (e x + d\right ) \sinh \left (e x + d\right ) + \sinh \left (e x + d\right )^{2} - 1}} {\left (\cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )\right )}}{b c \log \left (F\right )} \] Input:

integrate(F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2),x, alg 
orithm="fricas")
 

Output:

sqrt(2)*sqrt(f*sinh(e*x + d))*sqrt((g*cosh(e*x + d) + g*sinh(e*x + d))/(co 
sh(e*x + d)^2 + 2*cosh(e*x + d)*sinh(e*x + d) + sinh(e*x + d)^2 - 1))*(cos 
h((b*c*x + a*c)*log(F)) + sinh((b*c*x + a*c)*log(F)))/(b*c*log(F))
 

Sympy [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\text {Timed out} \] Input:

integrate(F**(c*(b*x+a))*(g*csch(e*x+d))**(1/2)*(f*sinh(e*x+d))**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\frac {F^{b c x} F^{a c} \sqrt {f} \sqrt {g}}{b c \log \left (F\right )} \] Input:

integrate(F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2),x, alg 
orithm="maxima")
 

Output:

F^(b*c*x)*F^(a*c)*sqrt(f)*sqrt(g)/(b*c*log(F))
 

Giac [F]

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

integrate(F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2),x, alg 
orithm="giac")
 

Output:

integrate(sqrt(g*csch(e*x + d))*sqrt(f*sinh(e*x + d))*F^((b*x + a)*c), x)
 

Mupad [B] (verification not implemented)

Time = 2.68 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.61 \[ \int F^{c (a+b x)} \sqrt {g \text {csch}(d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\frac {F^{a\,c+b\,c\,x}\,\sqrt {f\,\left (\frac {{\mathrm {e}}^{d+e\,x}}{2}-\frac {{\mathrm {e}}^{-d-e\,x}}{2}\right )}\,\sqrt {\frac {g}{\frac {{\mathrm {e}}^{d+e\,x}}{2}-\frac {{\mathrm {e}}^{-d-e\,x}}{2}}}}{b\,c\,\ln \left (F\right )} \] Input:

int(F^(c*(a + b*x))*(f*sinh(d + e*x))^(1/2)*(g/sinh(d + e*x))^(1/2),x)
 

Output:

(F^(a*c + b*c*x)*(f*(exp(d + e*x)/2 - exp(- d - e*x)/2))^(1/2)*(g/(exp(d + 
 e*x)/2 - exp(- d - e*x)/2))^(1/2))/(b*c*log(F))
 

Reduce [F]

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

int(F^(c*(b*x+a))*(g*csch(e*x+d))^(1/2)*(f*sinh(e*x+d))^(1/2),x)
 

Output:

sqrt(g)*f**((2*a*c + 1)/2)*int(f**(b*c*x)*sqrt(sinh(d + e*x))*sqrt(csch(d 
+ e*x)),x)