\(\int F^{c (a+b x)} (f \tan (d+e x))^p (g \tan (d+e x))^q \, dx\) [156]

Optimal result

Integrand size = 30, antiderivative size = 170 \[ \int F^{c (a+b x)} (f \tan (d+e x))^p (g \tan (d+e x))^q \, dx=\frac {i 2^{-1-p-q} \left (e^{2 i (d+e x)}\right )^{\frac {i b c \log (F)}{2 e}} \left (1-e^{2 i (d+e x)}\right ) \left (1+e^{2 i (d+e x)}\right )^{p+q} F^{c (a+b x)} \operatorname {AppellF1}\left (1+p+q,1+\frac {i b c \log (F)}{2 e},p+q,2+p+q,1-e^{2 i (d+e x)},\frac {1}{2} \left (1-e^{2 i (d+e x)}\right )\right ) (f \tan (d+e x))^p (g \tan (d+e x))^q}{e (1+p+q)} \] 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*tan(e*x+d))^p*(g* 
tan(e*x+d))^q/e/(1+p+q)
 

Mathematica [F]

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

Integrate[F^(c*(a + b*x))*(f*Tan[d + e*x])^p*(g*Tan[d + e*x])^q,x]
 

Output:

Integrate[F^(c*(a + b*x))*(f*Tan[d + e*x])^p*(g*Tan[d + e*x])^q, 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.

\(\displaystyle \int F^{c (a+b x)} (f \tan (d+e x))^p (g \tan (d+e x))^q \, dx\)

\(\Big \downarrow \) 2034

\(\displaystyle (f \tan (d+e x))^{-q} (g \tan (d+e x))^q \int F^{c (a+b x)} (f \tan (d+e x))^{p+q}dx\)

\(\Big \downarrow \) 7271

\(\displaystyle (f \tan (d+e x))^p (g \tan (d+e x))^q \tan ^{-p-q}(d+e x) \int F^{c (a+b x)} \tan ^{p+q}(d+e x)dx\)

\(\Big \downarrow \) 4967

\(\displaystyle (f \tan (d+e x))^p (g \tan (d+e x))^q \tan ^{-p-q}(d+e x) \int F^{a c+b x c} \tan ^{p+q}(d+e x)dx\)

\(\Big \downarrow \) 7299

\(\displaystyle (f \tan (d+e x))^p (g \tan (d+e x))^q \tan ^{-p-q}(d+e x) \int F^{a c+b x c} \tan ^{p+q}(d+e x)dx\)

Input:

Int[F^(c*(a + b*x))*(f*Tan[d + e*x])^p*(g*Tan[d + e*x])^q,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2034

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]
 

rule 4967

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]
 

rule 7271

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])
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (f \tan \left (e x +d \right )\right )^{p} \left (g \tan \left (e x +d \right )\right )^{q}d x\]

Input:

int(F^(c*(b*x+a))*(f*tan(e*x+d))^p*(g*tan(e*x+d))^q,x)
 

Output:

int(F^(c*(b*x+a))*(f*tan(e*x+d))^p*(g*tan(e*x+d))^q,x)
 

Fricas [F]

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

integrate(F^(c*(b*x+a))*(f*tan(e*x+d))^p*(g*tan(e*x+d))^q,x, algorithm="fr 
icas")
 

Output:

integral((f*tan(e*x + d))^p*(g*tan(e*x + d))^q*F^(b*c*x + a*c), x)
 

Sympy [F]

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

integrate(F**(c*(b*x+a))*(f*tan(e*x+d))**p*(g*tan(e*x+d))**q,x)
 

Output:

Integral(F**(c*(a + b*x))*(f*tan(d + e*x))**p*(g*tan(d + e*x))**q, x)
 

Maxima [F]

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

integrate(F^(c*(b*x+a))*(f*tan(e*x+d))^p*(g*tan(e*x+d))^q,x, algorithm="ma 
xima")
 

Output:

integrate((f*tan(e*x + d))^p*(g*tan(e*x + d))^q*F^((b*x + a)*c), x)
 

Giac [F]

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

integrate(F^(c*(b*x+a))*(f*tan(e*x+d))^p*(g*tan(e*x+d))^q,x, algorithm="gi 
ac")
 

Output:

integrate((f*tan(e*x + d))^p*(g*tan(e*x + d))^q*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

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

int(F^(c*(a + b*x))*(f*tan(d + e*x))^p*(g*tan(d + e*x))^q,x)
 

Output:

int(F^(c*(a + b*x))*(f*tan(d + e*x))^p*(g*tan(d + e*x))^q, x)
 

Reduce [F]

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

int(F^(c*(b*x+a))*(f*tan(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)**(p + q),x)