Integrand size = 30, antiderivative size = 181 \[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\frac {i 2^{-1+q} \left (e^{2 i (d+e x)}\right )^{\frac {1}{2} \left (-p+q+\frac {i b c \log (F)}{e}\right )} \left (1-e^{2 i (d+e x)}\right ) \left (1+e^{2 i (d+e x)}\right )^{-q} F^{c (a+b x)} \operatorname {AppellF1}\left (1-p,\frac {1}{2} \left (2-p+q+\frac {i b c \log (F)}{e}\right ),-q,2-p,1-e^{2 i (d+e x)},\frac {1}{2} \left (1-e^{2 i (d+e x)}\right )\right ) (g \cos (d+e x))^q (f \csc (d+e x))^p}{e (1-p)} \] Output:
I*2^(-1+q)*exp(2*I*(e*x+d))^(-1/2*p+1/2*q+1/2*I*b*c*ln(F)/e)*(1-exp(2*I*(e *x+d)))*F^(c*(b*x+a))*AppellF1(1-p,1-1/2*p+1/2*q+1/2*I*b*c*ln(F)/e,-q,2-p, 1-exp(2*I*(e*x+d)),1/2-1/2*exp(2*I*(e*x+d)))*(g*cos(e*x+d))^q*(f*csc(e*x+d ))^p/e/((1+exp(2*I*(e*x+d)))^q)/(1-p)
\[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx \] Input:
Integrate[F^(c*(a + b*x))*(g*Cos[d + e*x])^q*(f*Csc[d + e*x])^p,x]
Output:
Integrate[F^(c*(a + b*x))*(g*Cos[d + e*x])^q*(f*Csc[d + e*x])^p, x]
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.
\(\displaystyle \int F^{c (a+b x)} (f \csc (d+e x))^p (g \cos (d+e x))^q \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \cos ^{-q}(d+e x) (g \cos (d+e x))^q \int F^{c (a+b x)} \cos ^q(d+e x) (f \csc (d+e x))^pdx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^{-p}(d+e x) \cos ^{-q}(d+e x) (f \csc (d+e x))^p (g \cos (d+e x))^q \int F^{c (a+b x)} \cos ^q(d+e x) \csc ^p(d+e x)dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \csc ^{-p}(d+e x) \cos ^{-q}(d+e x) (f \csc (d+e x))^p (g \cos (d+e x))^q \int F^{a c+b x c} \cos ^q(d+e x) \csc ^p(d+e x)dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \csc ^{-p}(d+e x) \cos ^{-q}(d+e x) (f \csc (d+e x))^p (g \cos (d+e x))^q \int F^{a c+b x c} \cos ^q(d+e x) \csc ^p(d+e x)dx\) |
Input:
Int[F^(c*(a + b*x))*(g*Cos[d + e*x])^q*(f*Csc[d + e*x])^p,x]
Output:
$Aborted
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
\[\int F^{c \left (b x +a \right )} \left (g \cos \left (e x +d \right )\right )^{q} \left (f \csc \left (e x +d \right )\right )^{p}d x\]
Input:
int(F^(c*(b*x+a))*(g*cos(e*x+d))^q*(f*csc(e*x+d))^p,x)
Output:
int(F^(c*(b*x+a))*(g*cos(e*x+d))^q*(f*csc(e*x+d))^p,x)
\[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\int { \left (g \cos \left (e x + d\right )\right )^{q} \left (f \csc \left (e x + d\right )\right )^{p} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cos(e*x+d))^q*(f*csc(e*x+d))^p,x, algorithm="fr icas")
Output:
integral((g*cos(e*x + d))^q*(f*csc(e*x + d))^p*F^(b*c*x + a*c), x)
Timed out. \[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\text {Timed out} \] Input:
integrate(F**(c*(b*x+a))*(g*cos(e*x+d))**q*(f*csc(e*x+d))**p,x)
Output:
Timed out
\[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\int { \left (g \cos \left (e x + d\right )\right )^{q} \left (f \csc \left (e x + d\right )\right )^{p} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cos(e*x+d))^q*(f*csc(e*x+d))^p,x, algorithm="ma xima")
Output:
integrate((g*cos(e*x + d))^q*(f*csc(e*x + d))^p*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\int { \left (g \cos \left (e x + d\right )\right )^{q} \left (f \csc \left (e x + d\right )\right )^{p} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cos(e*x+d))^q*(f*csc(e*x+d))^p,x, algorithm="gi ac")
Output:
integrate((g*cos(e*x + d))^q*(f*csc(e*x + d))^p*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (g\,\cos \left (d+e\,x\right )\right )}^q\,{\left (\frac {f}{\sin \left (d+e\,x\right )}\right )}^p \,d x \] Input:
int(F^(c*(a + b*x))*(g*cos(d + e*x))^q*(f/sin(d + e*x))^p,x)
Output:
int(F^(c*(a + b*x))*(g*cos(d + e*x))^q*(f/sin(d + e*x))^p, x)
\[ \int F^{c (a+b x)} (g \cos (d+e x))^q (f \csc (d+e x))^p \, dx=g^{q} f^{a c +p} \left (\int f^{b c x} \csc \left (e x +d \right )^{p} \cos \left (e x +d \right )^{q}d x \right ) \] Input:
int(F^(c*(b*x+a))*(g*cos(e*x+d))^q*(f*csc(e*x+d))^p,x)
Output:
g**q*f**(a*c + p)*int(f**(b*c*x)*csc(d + e*x)**p*cos(d + e*x)**q,x)