\(\int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx\) [27]

Optimal result

Integrand size = 29, antiderivative size = 109 \[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\frac {8 e^{3 i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),\frac {1}{2} \left (5-\frac {i b c \log (F)}{e}\right ),\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}\right )}{(f-i g)^3 (3 e-i b c \log (F))} \] Output:

8*exp(3*I*(e*x+d))*F^(c*(b*x+a))*hypergeom([3, 3/2-1/2*I*b*c*ln(F)/e],[5/2 
-1/2*I*b*c*ln(F)/e],exp(2*I*(e*x+d))*(f+I*g)/(f-I*g))/(f-I*g)^3/(3*e-I*b*c 
*ln(F))
 

Mathematica [F]

\[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx \] Input:

Integrate[F^(c*(a + b*x))/(g*Cos[d + e*x] + f*Sin[d + e*x])^3,x]
 

Output:

Integrate[F^(c*(a + b*x))/(g*Cos[d + e*x] + f*Sin[d + e*x])^3, 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.

\(\displaystyle \int \frac {F^{c (a+b x)}}{(f \sin (d+e x)+g \cos (d+e x))^3} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {F^{a c+b c x}}{(f \sin (d+e x)+g \cos (d+e x))^3}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {F^{a c+b c x}}{(f \sin (d+e x)+g \cos (d+e x))^3}dx\)

Input:

Int[F^(c*(a + b*x))/(g*Cos[d + e*x] + f*Sin[d + e*x])^3,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]

\[\int \frac {F^{c \left (b x +a \right )}}{\left (g \cos \left (e x +d \right )+f \sin \left (e x +d \right )\right )^{3}}d x\]

Input:

int(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))^3,x)
 

Output:

int(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))^3,x)
 

Fricas [F]

\[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (g \cos \left (e x + d\right ) + f \sin \left (e x + d\right )\right )}^{3}} \,d x } \] Input:

integrate(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))^3,x, algorithm="fricas 
")
 

Output:

integral(F^(b*c*x + a*c)/(3*f^2*g*cos(e*x + d) - (3*f^2*g - g^3)*cos(e*x + 
 d)^3 + (f^3 - (f^3 - 3*f*g^2)*cos(e*x + d)^2)*sin(e*x + d)), x)
 

Sympy [F]

\[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\int \frac {F^{c \left (a + b x\right )}}{\left (f \sin {\left (d + e x \right )} + g \cos {\left (d + e x \right )}\right )^{3}}\, dx \] Input:

integrate(F**(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))**3,x)
 

Output:

Integral(F**(c*(a + b*x))/(f*sin(d + e*x) + g*cos(d + e*x))**3, x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\text {Timed out} \] Input:

integrate(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))^3,x, algorithm="maxima 
")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (g \cos \left (e x + d\right ) + f \sin \left (e x + d\right )\right )}^{3}} \,d x } \] Input:

integrate(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))^3,x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)/(g*cos(e*x + d) + f*sin(e*x + d))^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (g\,\cos \left (d+e\,x\right )+f\,\sin \left (d+e\,x\right )\right )}^3} \,d x \] Input:

int(F^(c*(a + b*x))/(g*cos(d + e*x) + f*sin(d + e*x))^3,x)
 

Output:

int(F^(c*(a + b*x))/(g*cos(d + e*x) + f*sin(d + e*x))^3, x)
 

Reduce [F]

\[ \int \frac {F^{c (a+b x)}}{(g \cos (d+e x)+f \sin (d+e x))^3} \, dx=f^{a c} \left (\int \frac {f^{b c x}}{\cos \left (e x +d \right )^{3} g^{3}+3 \cos \left (e x +d \right )^{2} \sin \left (e x +d \right ) f \,g^{2}+3 \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2} f^{2} g +\sin \left (e x +d \right )^{3} f^{3}}d x \right ) \] Input:

int(F^(c*(b*x+a))/(g*cos(e*x+d)+f*sin(e*x+d))^3,x)
 

Output:

f**(a*c)*int(f**(b*c*x)/(cos(d + e*x)**3*g**3 + 3*cos(d + e*x)**2*sin(d + 
e*x)*f*g**2 + 3*cos(d + e*x)*sin(d + e*x)**2*f**2*g + sin(d + e*x)**3*f**3 
),x)