\(\int \frac {F^{c (a+b x)}}{g \cos (d+e x)+f \sin (d+e x)} \, dx\) [25]

Optimal result

Integrand size = 29, antiderivative size = 107 \[ \int \frac {F^{c (a+b x)}}{g \cos (d+e x)+f \sin (d+e x)} \, dx=-\frac {2 e^{i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}\right )}{(f-i g) (e-i b c \log (F))} \] Output:

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

Mathematica [F]

\[ \int \frac {F^{c (a+b x)}}{g \cos (d+e x)+f \sin (d+e x)} \, dx=\int \frac {F^{c (a+b x)}}{g \cos (d+e x)+f \sin (d+e x)} \, dx \] Input:

Integrate[F^(c*(a + b*x))/(g*Cos[d + e*x] + f*Sin[d + e*x]),x]
 

Output:

Integrate[F^(c*(a + b*x))/(g*Cos[d + e*x] + f*Sin[d + e*x]), x]
 

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.

\(\displaystyle \int \frac {F^{c (a+b x)}}{f \sin (d+e x)+g \cos (d+e x)} \, dx\)

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

Int[F^(c*(a + b*x))/(g*Cos[d + e*x] + f*Sin[d + e*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \frac {F^{c \left (b x +a \right )}}{g \cos \left (e x +d \right )+f \sin \left (e x +d \right )}d x\]

Input:

int(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x)
 

Output:

int(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x)
 

Fricas [F]

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

integrate(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x, algorithm="fricas")
 

Output:

integral(F^(b*c*x + a*c)/(g*cos(e*x + d) + f*sin(e*x + d)), x)
 

Sympy [F]

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

integrate(F**(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x)
 

Output:

Integral(F**(c*(a + b*x))/(f*sin(d + e*x) + g*cos(d + e*x)), x)
 

Maxima [F]

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

integrate(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x, algorithm="maxima")
 

Output:

2*((F^(a*c)*b*c*g*log(F) + F^(a*c)*e*f)*F^(b*c*x)*cos(e*x + d) + (F^(a*c)* 
b*c*f*log(F) - F^(a*c)*e*g)*F^(b*c*x)*sin(e*x + d) + ((F^(a*c)*b*c*g*log(F 
) - F^(a*c)*e*f)*F^(b*c*x)*cos(e*x + d) - (F^(a*c)*b*c*f*log(F) + F^(a*c)* 
e*g)*F^(b*c*x)*sin(e*x + d))*cos(2*e*x + 2*d) + 2*(4*(F^(a*c)*b^2*c^2*e*lo 
g(F)^2 + F^(a*c)*e^3)*f*g^2*sin(2*e*x + 2*d) + (F^(a*c)*b^2*c^2*e*log(F)^2 
 + F^(a*c)*e^3)*f^2*g + (F^(a*c)*b^2*c^2*e*log(F)^2 + F^(a*c)*e^3)*g^3 + ( 
(F^(a*c)*b^2*c^2*e*log(F)^2 + F^(a*c)*e^3)*f^2*g + (F^(a*c)*b^2*c^2*e*log( 
F)^2 + F^(a*c)*e^3)*g^3)*cos(2*e*x + 2*d)^2 + ((F^(a*c)*b^2*c^2*e*log(F)^2 
 + F^(a*c)*e^3)*f^2*g + (F^(a*c)*b^2*c^2*e*log(F)^2 + F^(a*c)*e^3)*g^3)*si 
n(2*e*x + 2*d)^2 - 2*((F^(a*c)*b^2*c^2*e*log(F)^2 + F^(a*c)*e^3)*f^2*g - ( 
F^(a*c)*b^2*c^2*e*log(F)^2 + F^(a*c)*e^3)*g^3)*cos(2*e*x + 2*d))*integrate 
(-((b*c*f^2*log(F) - b*c*g^2*log(F) - 2*e*f*g)*F^(b*c*x)*cos(e*x + d) - (2 
*b*c*f*g*log(F) + e*f^2 - e*g^2)*F^(b*c*x)*sin(e*x + d) + ((b*c*f^2*log(F) 
 - b*c*g^2*log(F) + 2*e*f*g)*F^(b*c*x)*cos(e*x + d) + (2*b*c*f*g*log(F) - 
e*f^2 + e*g^2)*F^(b*c*x)*sin(e*x + d))*cos(4*e*x + 4*d) - 2*((b*c*f^2*log( 
F) + b*c*g^2*log(F))*F^(b*c*x)*cos(e*x + d) - (e*f^2 + e*g^2)*F^(b*c*x)*si 
n(e*x + d))*cos(2*e*x + 2*d) - ((2*b*c*f*g*log(F) - e*f^2 + e*g^2)*F^(b*c* 
x)*cos(e*x + d) - (b*c*f^2*log(F) - b*c*g^2*log(F) + 2*e*f*g)*F^(b*c*x)*si 
n(e*x + d))*sin(4*e*x + 4*d) - 2*((e*f^2 + e*g^2)*F^(b*c*x)*cos(e*x + d) + 
 (b*c*f^2*log(F) + b*c*g^2*log(F))*F^(b*c*x)*sin(e*x + d))*sin(2*e*x + ...
 

Giac [F]

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

integrate(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)/(g*cos(e*x + d) + f*sin(e*x + d)), x)
 

Mupad [F(-1)]

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

int(F^(c*(a + b*x))/(g*cos(d + e*x) + f*sin(d + e*x)),x)
 

Output:

int(F^(c*(a + b*x))/(g*cos(d + e*x) + f*sin(d + e*x)), x)
 

Reduce [F]

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

int(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d)),x)
 

Output:

f**(a*c)*int(f**(b*c*x)/(cos(d + e*x)*g + sin(d + e*x)*f),x)