\(\int F^{c (a+b x)} \sqrt {f \csc (d+e x)} \sqrt {g \csc (d+e x)} \, dx\) [131]

Optimal result

Integrand size = 34, antiderivative size = 106 \[ \int F^{c (a+b x)} \sqrt {f \csc (d+e x)} \sqrt {g \csc (d+e x)} \, dx=-\frac {2 e^{i (d+e x)} f F^{c (a+b x)} \sqrt {g \csc (d+e x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{\sqrt {f \csc (d+e x)} (e-i b c \log (F))} \] Output:

-2*exp(I*(e*x+d))*f*F^(c*(b*x+a))*(g*csc(e*x+d))^(1/2)*hypergeom([1, 1/2*( 
e-I*b*c*ln(F))/e],[3/2-1/2*I*b*c*ln(F)/e],exp(2*I*(e*x+d)))/(f*csc(e*x+d)) 
^(1/2)/(e-I*b*c*ln(F))
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.97 \[ \int F^{c (a+b x)} \sqrt {f \csc (d+e x)} \sqrt {g \csc (d+e x)} \, dx=\frac {f F^{c (a+b x)} \csc (d) \sqrt {g \csc (d+e x)}}{b c \sqrt {f \csc (d+e x)} \log (F)}+\frac {i f F^{c (a+b x)} \csc \left (\frac {d}{2}\right ) \sqrt {g \csc (d+e x)} \sec \left (\frac {d}{2}\right ) \left (i+\operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{e},1-\frac {i b c \log (F)}{e},-\cos (d+e x)-i \sin (d+e x)\right ) \sin (d)-\operatorname {Hypergeometric2F1}\left (1,-\frac {i b c \log (F)}{e},1-\frac {i b c \log (F)}{e},\cos (d+e x)+i \sin (d+e x)\right ) \sin (d)\right )}{2 b c \sqrt {f \csc (d+e x)} \log (F)} \] Input:

Integrate[F^(c*(a + b*x))*Sqrt[f*Csc[d + e*x]]*Sqrt[g*Csc[d + e*x]],x]
 

Output:

(f*F^(c*(a + b*x))*Csc[d]*Sqrt[g*Csc[d + e*x]])/(b*c*Sqrt[f*Csc[d + e*x]]* 
Log[F]) + ((I/2)*f*F^(c*(a + b*x))*Csc[d/2]*Sqrt[g*Csc[d + e*x]]*Sec[d/2]* 
(I + Hypergeometric2F1[1, ((-I)*b*c*Log[F])/e, 1 - (I*b*c*Log[F])/e, -Cos[ 
d + e*x] - I*Sin[d + e*x]]*Sin[d] - Hypergeometric2F1[1, ((-I)*b*c*Log[F]) 
/e, 1 - (I*b*c*Log[F])/e, Cos[d + e*x] + I*Sin[d + e*x]]*Sin[d]))/(b*c*Sqr 
t[f*Csc[d + e*x]]*Log[F])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2031, 4953}

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.

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

\(\Big \downarrow \) 2031

\(\displaystyle \frac {f \sqrt {g \csc (d+e x)} \int F^{c (a+b x)} \csc (d+e x)dx}{\sqrt {f \csc (d+e x)}}\)

\(\Big \downarrow \) 4953

\(\displaystyle -\frac {2 f e^{i (d+e x)} F^{c (a+b x)} \sqrt {g \csc (d+e x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{(e-i b c \log (F)) \sqrt {f \csc (d+e x)}}\)

Input:

Int[F^(c*(a + b*x))*Sqrt[f*Csc[d + e*x]]*Sqrt[g*Csc[d + e*x]],x]
 

Output:

(-2*E^(I*(d + e*x))*f*F^(c*(a + b*x))*Sqrt[g*Csc[d + e*x]]*Hypergeometric2 
F1[1, (e - I*b*c*Log[F])/(2*e), (3 - (I*b*c*Log[F])/e)/2, E^((2*I)*(d + e* 
x))])/(Sqrt[f*Csc[d + e*x]]*(e - I*b*c*Log[F]))
 

Defintions of rubi rules used

rule 2031

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + 1/ 
2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v])   Int[v^(m + n)*Fx, x], x] /; FreeQ[{a 
, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
 

rule 4953

Int[Csc[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symb 
ol] :> Simp[(-2*I)^n*E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F] 
))*Hypergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(Log[F] 
/(2*e)), E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ 
[n]
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \sqrt {f \csc \left (e x +d \right )}\, \sqrt {g \csc \left (e x +d \right )}d x\]

Input:

int(F^(c*(b*x+a))*(f*csc(e*x+d))^(1/2)*(g*csc(e*x+d))^(1/2),x)
 

Output:

int(F^(c*(b*x+a))*(f*csc(e*x+d))^(1/2)*(g*csc(e*x+d))^(1/2),x)
 

Fricas [F]

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

integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^(1/2)*(g*csc(e*x+d))^(1/2),x, algor 
ithm="fricas")
 

Output:

integral(sqrt(f*csc(e*x + d))*sqrt(g*csc(e*x + d))*F^(b*c*x + a*c), x)
 

Sympy [F]

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

integrate(F**(c*(b*x+a))*(f*csc(e*x+d))**(1/2)*(g*csc(e*x+d))**(1/2),x)
 

Output:

Integral(F**(c*(a + b*x))*sqrt(f*csc(d + e*x))*sqrt(g*csc(d + e*x)), x)
 

Maxima [F]

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

integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^(1/2)*(g*csc(e*x+d))^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate(sqrt(f*csc(e*x + d))*sqrt(g*csc(e*x + d))*F^((b*x + a)*c), x)
 

Giac [F]

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

integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^(1/2)*(g*csc(e*x+d))^(1/2),x, algor 
ithm="giac")
 

Output:

integrate(sqrt(f*csc(e*x + d))*sqrt(g*csc(e*x + d))*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \sqrt {f \csc (d+e x)} \sqrt {g \csc (d+e x)} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {\frac {f}{\sin \left (d+e\,x\right )}}\,\sqrt {\frac {g}{\sin \left (d+e\,x\right )}} \,d x \] Input:

int(F^(c*(a + b*x))*(f/sin(d + e*x))^(1/2)*(g/sin(d + e*x))^(1/2),x)
 

Output:

int(F^(c*(a + b*x))*(f/sin(d + e*x))^(1/2)*(g/sin(d + e*x))^(1/2), x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \sqrt {f \csc (d+e x)} \sqrt {g \csc (d+e x)} \, dx=\sqrt {g}\, f^{a c +\frac {1}{2}} \left (\int f^{b c x} \csc \left (e x +d \right )d x \right ) \] Input:

int(F^(c*(b*x+a))*(f*csc(e*x+d))^(1/2)*(g*csc(e*x+d))^(1/2),x)
 

Output:

sqrt(g)*f**((2*a*c + 1)/2)*int(f**(b*c*x)*csc(d + e*x),x)