Integrand size = 34, antiderivative size = 119 \[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=-\frac {2 F^{c (a+b x)} \sqrt {g \cot (d+e x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {-e-2 i b c \log (F)}{4 e},\frac {1}{4} \left (3-\frac {2 i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right ) \sqrt {f \sin (d+e x)}}{\sqrt {1+e^{2 i (d+e x)}} (i e-2 b c \log (F))} \] Output:
-2*F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*hypergeom([-1/2, 1/4*(-e-2*I*b*c*ln( F))/e],[3/4-1/2*I*b*c*ln(F)/e],-exp(2*I*(e*x+d)))*(f*sin(e*x+d))^(1/2)/(1+ exp(2*I*(e*x+d)))^(1/2)/(I*e-2*b*c*ln(F))
Time = 2.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10 \[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\frac {4 i F^{c (a+b x)} \cos (d+e x) \sqrt {g \cot (d+e x)} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4}-\frac {i b c \log (F)}{2 e},\frac {3}{4}-\frac {i b c \log (F)}{2 e},-\cos (2 (d+e x))-i \sin (2 (d+e x))\right ) (\cos (d+e x)+i \sin (d+e x)) \sqrt {f \sin (d+e x)}}{e+2 i b c \log (F)} \] Input:
Integrate[F^(c*(a + b*x))*Sqrt[g*Cot[d + e*x]]*Sqrt[f*Sin[d + e*x]],x]
Output:
((4*I)*F^(c*(a + b*x))*Cos[d + e*x]*Sqrt[g*Cot[d + e*x]]*Hypergeometric2F1 [1, 5/4 - ((I/2)*b*c*Log[F])/e, 3/4 - ((I/2)*b*c*Log[F])/e, -Cos[2*(d + e* x)] - I*Sin[2*(d + e*x)]]*(Cos[d + e*x] + I*Sin[d + e*x])*Sqrt[f*Sin[d + e *x]])/(e + (2*I)*b*c*Log[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.
\(\displaystyle \int F^{c (a+b x)} \sqrt {f \sin (d+e x)} \sqrt {g \cot (d+e x)} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \frac {\sqrt {g \cot (d+e x)} \int F^{c (a+b x)} \sqrt {\cot (d+e x)} \sqrt {f \sin (d+e x)}dx}{\sqrt {\cot (d+e x)}}\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \frac {\sqrt {f \sin (d+e x)} \sqrt {g \cot (d+e x)} \int F^{c (a+b x)} \sqrt {\cot (d+e x)} \sqrt {\sin (d+e x)}dx}{\sqrt {\sin (d+e x)} \sqrt {\cot (d+e x)}}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {\sqrt {f \sin (d+e x)} \sqrt {g \cot (d+e x)} \int F^{a c+b x c} \sqrt {\cot (d+e x)} \sqrt {\sin (d+e x)}dx}{\sqrt {\sin (d+e x)} \sqrt {\cot (d+e x)}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {\sqrt {f \sin (d+e x)} \sqrt {g \cot (d+e x)} \int F^{a c+b x c} \sqrt {\cot (d+e x)} \sqrt {\sin (d+e x)}dx}{\sqrt {\sin (d+e x)} \sqrt {\cot (d+e x)}}\) |
Input:
Int[F^(c*(a + b*x))*Sqrt[g*Cot[d + e*x]]*Sqrt[f*Sin[d + e*x]],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])
\[\int F^{c \left (b x +a \right )} \sqrt {g \cot \left (e x +d \right )}\, \sqrt {f \sin \left (e x +d \right )}d x\]
Input:
int(F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*(f*sin(e*x+d))^(1/2),x)
Output:
int(F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*(f*sin(e*x+d))^(1/2),x)
\[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\int { \sqrt {g \cot \left (e x + d\right )} \sqrt {f \sin \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*(f*sin(e*x+d))^(1/2),x, algor ithm="fricas")
Output:
integral(sqrt(g*cot(e*x + d))*sqrt(f*sin(e*x + d))*F^(b*c*x + a*c), x)
Timed out. \[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\text {Timed out} \] Input:
integrate(F**(c*(b*x+a))*(g*cot(e*x+d))**(1/2)*(f*sin(e*x+d))**(1/2),x)
Output:
Timed out
\[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\int { \sqrt {g \cot \left (e x + d\right )} \sqrt {f \sin \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*(f*sin(e*x+d))^(1/2),x, algor ithm="maxima")
Output:
integrate(sqrt(g*cot(e*x + d))*sqrt(f*sin(e*x + d))*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\int { \sqrt {g \cot \left (e x + d\right )} \sqrt {f \sin \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*(f*sin(e*x+d))^(1/2),x, algor ithm="giac")
Output:
integrate(sqrt(g*cot(e*x + d))*sqrt(f*sin(e*x + d))*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {g\,\mathrm {cot}\left (d+e\,x\right )}\,\sqrt {f\,\sin \left (d+e\,x\right )} \,d x \] Input:
int(F^(c*(a + b*x))*(g*cot(d + e*x))^(1/2)*(f*sin(d + e*x))^(1/2),x)
Output:
int(F^(c*(a + b*x))*(g*cot(d + e*x))^(1/2)*(f*sin(d + e*x))^(1/2), x)
\[ \int F^{c (a+b x)} \sqrt {g \cot (d+e x)} \sqrt {f \sin (d+e x)} \, dx=\sqrt {g}\, f^{a c +\frac {1}{2}} \left (\int f^{b c x} \sqrt {\sin \left (e x +d \right )}\, \sqrt {\cot \left (e x +d \right )}d x \right ) \] Input:
int(F^(c*(b*x+a))*(g*cot(e*x+d))^(1/2)*(f*sin(e*x+d))^(1/2),x)
Output:
sqrt(g)*f**((2*a*c + 1)/2)*int(f**(b*c*x)*sqrt(sin(d + e*x))*sqrt(cot(d + e*x)),x)