Integrand size = 30, antiderivative size = 180 \[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\frac {i 2^{-1-q} \left (e^{2 i (d+e x)}\right )^{\frac {1}{2} \left (-p+\frac {i b c \log (F)}{e}\right )} \left (1-e^{2 i (d+e x)}\right ) \left (1+e^{2 i (d+e x)}\right )^q F^{c (a+b x)} \operatorname {AppellF1}\left (1-p+q,\frac {1}{2} \left (2-p+\frac {i b c \log (F)}{e}\right ),q,2-p+q,1-e^{2 i (d+e x)},\frac {1}{2} \left (1-e^{2 i (d+e x)}\right )\right ) (f \csc (d+e x))^p (g \tan (d+e x))^q}{e (1-p+q)} \] Output:
I*2^(-1-q)*exp(2*I*(e*x+d))^(-1/2*p+1/2*I*b*c*ln(F)/e)*(1-exp(2*I*(e*x+d)) )*(1+exp(2*I*(e*x+d)))^q*F^(c*(b*x+a))*AppellF1(1-p+q,1-1/2*p+1/2*I*b*c*ln (F)/e,q,2-p+q,1-exp(2*I*(e*x+d)),1/2-1/2*exp(2*I*(e*x+d)))*(f*csc(e*x+d))^ p*(g*tan(e*x+d))^q/e/(1-p+q)
\[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx \] Input:
Integrate[F^(c*(a + b*x))*(f*Csc[d + e*x])^p*(g*Tan[d + e*x])^q,x]
Output:
Integrate[F^(c*(a + b*x))*(f*Csc[d + e*x])^p*(g*Tan[d + e*x])^q, 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.
\(\displaystyle \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^{-p}(d+e x) (f \csc (d+e x))^p \int F^{c (a+b x)} \csc ^p(d+e x) (g \tan (d+e x))^qdx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^{-p}(d+e x) \tan ^{-q}(d+e x) (f \csc (d+e x))^p (g \tan (d+e x))^q \int F^{c (a+b x)} \csc ^p(d+e x) \tan ^q(d+e x)dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \csc ^{-p}(d+e x) \tan ^{-q}(d+e x) (f \csc (d+e x))^p (g \tan (d+e x))^q \int F^{a c+b x c} \csc ^p(d+e x) \tan ^q(d+e x)dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \csc ^{-p}(d+e x) \tan ^{-q}(d+e x) (f \csc (d+e x))^p (g \tan (d+e x))^q \int F^{a c+b x c} \csc ^p(d+e x) \tan ^q(d+e x)dx\) |
Input:
Int[F^(c*(a + b*x))*(f*Csc[d + e*x])^p*(g*Tan[d + e*x])^q,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])
\[\int F^{c \left (b x +a \right )} \left (f \csc \left (e x +d \right )\right )^{p} \left (g \tan \left (e x +d \right )\right )^{q}d x\]
Input:
int(F^(c*(b*x+a))*(f*csc(e*x+d))^p*(g*tan(e*x+d))^q,x)
Output:
int(F^(c*(b*x+a))*(f*csc(e*x+d))^p*(g*tan(e*x+d))^q,x)
\[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\int { \left (f \csc \left (e x + d\right )\right )^{p} \left (g \tan \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^p*(g*tan(e*x+d))^q,x, algorithm="fr icas")
Output:
integral((f*csc(e*x + d))^p*(g*tan(e*x + d))^q*F^(b*c*x + a*c), x)
\[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\int F^{c \left (a + b x\right )} \left (f \csc {\left (d + e x \right )}\right )^{p} \left (g \tan {\left (d + e x \right )}\right )^{q}\, dx \] Input:
integrate(F**(c*(b*x+a))*(f*csc(e*x+d))**p*(g*tan(e*x+d))**q,x)
Output:
Integral(F**(c*(a + b*x))*(f*csc(d + e*x))**p*(g*tan(d + e*x))**q, x)
\[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\int { \left (f \csc \left (e x + d\right )\right )^{p} \left (g \tan \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^p*(g*tan(e*x+d))^q,x, algorithm="ma xima")
Output:
integrate((f*csc(e*x + d))^p*(g*tan(e*x + d))^q*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\int { \left (f \csc \left (e x + d\right )\right )^{p} \left (g \tan \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^p*(g*tan(e*x+d))^q,x, algorithm="gi ac")
Output:
integrate((f*csc(e*x + d))^p*(g*tan(e*x + d))^q*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (g\,\mathrm {tan}\left (d+e\,x\right )\right )}^q\,{\left (\frac {f}{\sin \left (d+e\,x\right )}\right )}^p \,d x \] Input:
int(F^(c*(a + b*x))*(g*tan(d + e*x))^q*(f/sin(d + e*x))^p,x)
Output:
int(F^(c*(a + b*x))*(g*tan(d + e*x))^q*(f/sin(d + e*x))^p, x)
\[ \int F^{c (a+b x)} (f \csc (d+e x))^p (g \tan (d+e x))^q \, dx=g^{q} f^{a c +p} \left (\int f^{b c x} \tan \left (e x +d \right )^{q} \csc \left (e x +d \right )^{p}d x \right ) \] Input:
int(F^(c*(b*x+a))*(f*csc(e*x+d))^p*(g*tan(e*x+d))^q,x)
Output:
g**q*f**(a*c + p)*int(f**(b*c*x)*tan(d + e*x)**q*csc(d + e*x)**p,x)