Integrand size = 31, antiderivative size = 154 \[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\frac {2 F^{c (a+b x)} \left (f-g-e^{2 d+2 e x} (f+g)\right ) \sqrt {1-\frac {e^{2 d+2 e x} (f+g)}{f-g}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3+\frac {2 b c \log (F)}{e}\right ),\frac {1}{4} \left (7+\frac {2 b c \log (F)}{e}\right ),\frac {e^{2 d+2 e x} (f+g)}{f-g}\right )}{(f-g) (3 e+2 b c \log (F)) (g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \] Output:
2*F^(c*(b*x+a))*(f-g-exp(2*e*x+2*d)*(f+g))*(1-exp(2*e*x+2*d)*(f+g)/(f-g))^ (1/2)*hypergeom([3/2, 3/4+1/2*b*c*ln(F)/e],[7/4+1/2*b*c*ln(F)/e],exp(2*e*x +2*d)*(f+g)/(f-g))/(f-g)/(2*b*c*ln(F)+3*e)/(g*cosh(e*x+d)+f*sinh(e*x+d))^( 3/2)
Time = 5.24 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.97 \[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\frac {\sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \left (-\frac {2 e F^{a c+b c x}}{e f g+2 b c g^2 \log (F)}+\frac {2 F^{c (a+b x)} \sinh (d+e x)}{g (g \cosh (d+e x)+f \sinh (d+e x))}-\frac {2 F^{c (a+b x)} (e+2 b c \log (F)) \left (\operatorname {Hypergeometric2F1}\left (1,\frac {e+2 b c \log (F)}{4 e},\frac {3}{4}+\frac {b c \log (F)}{2 e},\frac {(f+g) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))}{f-g}\right ) (3 e+2 b c \log (F))+\operatorname {Hypergeometric2F1}\left (1,\frac {5}{4}+\frac {b c \log (F)}{2 e},\frac {7}{4}+\frac {b c \log (F)}{2 e},\frac {(f+g) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))}{f-g}\right ) (e-2 b c \log (F)) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))\right )}{(f-g) (3 e+2 b c \log (F)) (e f+2 b c g \log (F))}\right )}{e} \] Input:
Integrate[F^(c*(a + b*x))/(g*Cosh[d + e*x] + f*Sinh[d + e*x])^(3/2),x]
Output:
(Sqrt[g*Cosh[d + e*x] + f*Sinh[d + e*x]]*((-2*e*F^(a*c + b*c*x))/(e*f*g + 2*b*c*g^2*Log[F]) + (2*F^(c*(a + b*x))*Sinh[d + e*x])/(g*(g*Cosh[d + e*x] + f*Sinh[d + e*x])) - (2*F^(c*(a + b*x))*(e + 2*b*c*Log[F])*(Hypergeometri c2F1[1, (e + 2*b*c*Log[F])/(4*e), 3/4 + (b*c*Log[F])/(2*e), ((f + g)*(Cosh [2*(d + e*x)] + Sinh[2*(d + e*x)]))/(f - g)]*(3*e + 2*b*c*Log[F]) + Hyperg eometric2F1[1, 5/4 + (b*c*Log[F])/(2*e), 7/4 + (b*c*Log[F])/(2*e), ((f + g )*(Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)]))/(f - g)]*(e - 2*b*c*Log[F])*(Co sh[2*(d + e*x)] + Sinh[2*(d + e*x)])))/((f - g)*(3*e + 2*b*c*Log[F])*(e*f + 2*b*c*g*Log[F]))))/e
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.
| \(\displaystyle \int \frac {F^{c (a+b x)}}{(f \sinh (d+e x)+g \cosh (d+e x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {F^{a c+b c x}}{(f \sinh (d+e x)+g \cosh (d+e x))^{3/2}}dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {F^{a c+b c x}}{(f \sinh (d+e x)+g \cosh (d+e x))^{3/2}}dx\) |
Input:
Int[F^(c*(a + b*x))/(g*Cosh[d + e*x] + f*Sinh[d + e*x])^(3/2),x]
Output:
$Aborted
\[\int \frac {F^{c \left (b x +a \right )}}{\left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{\frac {3}{2}}}d x\]
Input:
int(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x)
Output:
int(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x)
\[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x, algorithm=" fricas")
Output:
integral(sqrt(g*cosh(e*x + d) + f*sinh(e*x + d))*F^(b*c*x + a*c)/(g^2*cosh (e*x + d)^2 + 2*f*g*cosh(e*x + d)*sinh(e*x + d) + f^2*sinh(e*x + d)^2), x)
\[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\int \frac {F^{c \left (a + b x\right )}}{\left (f \sinh {\left (d + e x \right )} + g \cosh {\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(F**(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))**(3/2),x)
Output:
Integral(F**(c*(a + b*x))/(f*sinh(d + e*x) + g*cosh(d + e*x))**(3/2), x)
\[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x, algorithm=" maxima")
Output:
integrate(F^((b*x + a)*c)/(g*cosh(e*x + d) + f*sinh(e*x + d))^(3/2), x)
\[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x, algorithm=" giac")
Output:
integrate(F^((b*x + a)*c)/(g*cosh(e*x + d) + f*sinh(e*x + d))^(3/2), x)
Timed out. \[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (g\,\mathrm {cosh}\left (d+e\,x\right )+f\,\mathrm {sinh}\left (d+e\,x\right )\right )}^{3/2}} \,d x \] Input:
int(F^(c*(a + b*x))/(g*cosh(d + e*x) + f*sinh(d + e*x))^(3/2),x)
Output:
int(F^(c*(a + b*x))/(g*cosh(d + e*x) + f*sinh(d + e*x))^(3/2), x)
\[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^{3/2}} \, dx=f^{a c} \left (\int \frac {f^{b c x} \sqrt {\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f}}{\cosh \left (e x +d \right )^{2} g^{2}+2 \cosh \left (e x +d \right ) \sinh \left (e x +d \right ) f g +\sinh \left (e x +d \right )^{2} f^{2}}d x \right ) \] Input:
int(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x)
Output:
f**(a*c)*int((f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f))/(cosh(d + e*x)**2*g**2 + 2*cosh(d + e*x)*sinh(d + e*x)*f*g + sinh(d + e*x)**2*f**2 ),x)