Integrand size = 30, antiderivative size = 186 \[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\frac {i 2^{-1+p+q} \left (e^{2 i (d+e x)}\right )^{\frac {i b c \log (F)}{2 e}} \left (1-e^{2 i (d+e x)}\right ) \left (1+e^{2 i (d+e x)}\right )^{-p-q} F^{c (a+b x)} \operatorname {AppellF1}\left (1-p-q,1+\frac {i b c \log (F)}{2 e},-p-q,2-p-q,1-e^{2 i (d+e x)},\frac {1}{2} \left (1-e^{2 i (d+e x)}\right )\right ) (f \cot (d+e x))^p (g \cot (d+e x))^q}{e (1-p-q)} \] Output:
I*2^(-1+p+q)*exp(2*I*(e*x+d))^(1/2*I*b*c*ln(F)/e)*(1-exp(2*I*(e*x+d)))*(1+ exp(2*I*(e*x+d)))^(-p-q)*F^(c*(b*x+a))*AppellF1(1-p-q,1+1/2*I*b*c*ln(F)/e, -p-q,2-p-q,1-exp(2*I*(e*x+d)),1/2-1/2*exp(2*I*(e*x+d)))*(f*cot(e*x+d))^p*( g*cot(e*x+d))^q/e/(1-p-q)
\[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx \] Input:
Integrate[F^(c*(a + b*x))*(f*Cot[d + e*x])^p*(g*Cot[d + e*x])^q,x]
Output:
Integrate[F^(c*(a + b*x))*(f*Cot[d + e*x])^p*(g*Cot[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.
\(\displaystyle \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx\) |
\(\Big \downarrow \) 2034 |
\(\displaystyle (f \cot (d+e x))^{-q} (g \cot (d+e x))^q \int F^{c (a+b x)} (f \cot (d+e x))^{p+q}dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle (f \cot (d+e x))^p (g \cot (d+e x))^q \cot ^{-p-q}(d+e x) \int F^{c (a+b x)} \cot ^{p+q}(d+e x)dx\) |
\(\Big \downarrow \) 4967 |
\(\displaystyle (f \cot (d+e x))^p (g \cot (d+e x))^q \cot ^{-p-q}(d+e x) \int F^{a c+b x c} \cot ^{p+q}(d+e x)dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle (f \cot (d+e x))^p (g \cot (d+e x))^q \cot ^{-p-q}(d+e x) \int F^{a c+b x c} \cot ^{p+q}(d+e x)dx\) |
Input:
Int[F^(c*(a + b*x))*(f*Cot[d + e*x])^p*(g*Cot[d + e*x])^q,x]
Output:
$Aborted
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart [n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])) Int[(a*v)^(m + n )*Fx, x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[m + n]
Int[(F_)^((c_.)*(u_))*(G_)[v_]^(n_.), x_Symbol] :> Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x] /; FreeQ[{F, c, n}, x] && TrigQ[G] && Linear Q[{u, v}, x] && !LinearMatchQ[{u, v}, x]
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])
\[\int F^{c \left (b x +a \right )} \left (f \cot \left (e x +d \right )\right )^{p} \left (g \cot \left (e x +d \right )\right )^{q}d x\]
Input:
int(F^(c*(b*x+a))*(f*cot(e*x+d))^p*(g*cot(e*x+d))^q,x)
Output:
int(F^(c*(b*x+a))*(f*cot(e*x+d))^p*(g*cot(e*x+d))^q,x)
\[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\int { \left (f \cot \left (e x + d\right )\right )^{p} \left (g \cot \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cot(e*x+d))^p*(g*cot(e*x+d))^q,x, algorithm="fr icas")
Output:
integral((f*cot(e*x + d))^p*(g*cot(e*x + d))^q*F^(b*c*x + a*c), x)
\[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\int F^{c \left (a + b x\right )} \left (f \cot {\left (d + e x \right )}\right )^{p} \left (g \cot {\left (d + e x \right )}\right )^{q}\, dx \] Input:
integrate(F**(c*(b*x+a))*(f*cot(e*x+d))**p*(g*cot(e*x+d))**q,x)
Output:
Integral(F**(c*(a + b*x))*(f*cot(d + e*x))**p*(g*cot(d + e*x))**q, x)
\[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\int { \left (f \cot \left (e x + d\right )\right )^{p} \left (g \cot \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cot(e*x+d))^p*(g*cot(e*x+d))^q,x, algorithm="ma xima")
Output:
integrate((f*cot(e*x + d))^p*(g*cot(e*x + d))^q*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\int { \left (f \cot \left (e x + d\right )\right )^{p} \left (g \cot \left (e x + d\right )\right )^{q} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cot(e*x+d))^p*(g*cot(e*x+d))^q,x, algorithm="gi ac")
Output:
integrate((f*cot(e*x + d))^p*(g*cot(e*x + d))^q*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (f\,\mathrm {cot}\left (d+e\,x\right )\right )}^p\,{\left (g\,\mathrm {cot}\left (d+e\,x\right )\right )}^q \,d x \] Input:
int(F^(c*(a + b*x))*(f*cot(d + e*x))^p*(g*cot(d + e*x))^q,x)
Output:
int(F^(c*(a + b*x))*(f*cot(d + e*x))^p*(g*cot(d + e*x))^q, x)
\[ \int F^{c (a+b x)} (f \cot (d+e x))^p (g \cot (d+e x))^q \, dx=g^{q} f^{a c +p} \left (\int f^{b c x} \cot \left (e x +d \right )^{p +q}d x \right ) \] Input:
int(F^(c*(b*x+a))*(f*cot(e*x+d))^p*(g*cot(e*x+d))^q,x)
Output:
g**q*f**(a*c + p)*int(f**(b*c*x)*cot(d + e*x)**(p + q),x)