\(\int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^n \, dx\) [139]

Optimal result

Integrand size = 29, antiderivative size = 159 \[ \int F^{c (a+b x)} (g \cos (d+e x)+f \sin (d+e x))^n \, dx=-\frac {F^{c (a+b x)} \left (\frac {i f-e^{2 i (d+e x)} (i f-g)+g}{i f+g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2-n-\frac {i b c \log (F)}{e}\right ),\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}\right ) (g \cos (d+e x)+f \sin (d+e x))^n}{i e n-b c \log (F)} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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