\(\int F^{c (a+b x)} \sqrt {g \cosh (d+e x)} \sqrt {f \sinh (d+e x)} \, dx\) [146]

Optimal result

Integrand size = 34, antiderivative size = 121 \[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)} \sqrt {f \sinh (d+e x)} \, dx=-\frac {F^{c (a+b x)} \sqrt {g \cosh (d+e x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {b c \log (F)}{e}\right ),\frac {1}{4} \left (3+\frac {b c \log (F)}{e}\right ),e^{4 d+4 e x}\right ) \sqrt {f \sinh (d+e x)}}{\sqrt {1-e^{2 d+2 e x}} \sqrt {1+e^{2 d+2 e x}} (e-b c \log (F))} \] Output:

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

Mathematica [A] (verified)

Time = 2.78 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)} \sqrt {f \sinh (d+e x)} \, dx=\frac {F^{c (a+b x)} \sqrt {g \cosh (d+e x)} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4}-\frac {b c \log (F)}{4 e};\frac {3}{4}-\frac {b c \log (F)}{4 e};e^{-4 (d+e x)}\right ) \sqrt {f \sinh (d+e x)}}{\sqrt {1-e^{-4 (d+e x)}} (e+b c \log (F))} \] Input:

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

Output:

(F^(c*(a + b*x))*Sqrt[g*Cosh[d + e*x]]*HypergeometricPFQ[{-1/2, -1/4 - (b* 
c*Log[F])/(4*e)}, {3/4 - (b*c*Log[F])/(4*e)}, E^(-4*(d + e*x))]*Sqrt[f*Sin 
h[d + e*x]])/(Sqrt[1 - E^(-4*(d + e*x))]*(e + 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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \sqrt {g \cosh \left (e x +d \right )}\, \sqrt {f \sinh \left (e x +d \right )}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(F**(c*(a + b*x))*sqrt(f*sinh(d + e*x))*sqrt(g*cosh(d + e*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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