\(\int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx\) [47]

Optimal result

Integrand size = 39, antiderivative size = 114 \[ \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {d+c \csc (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{(a+b) f \sqrt {c+d \sin (e+f x)}} \] Output:

2*(-cot(f*x+e)^2)^(1/2)*((d+c*csc(f*x+e))/(c+d))^(1/2)*EllipticPi(1/2*(1-c 
sc(f*x+e))^(1/2)*2^(1/2),2*a/(a+b),2^(1/2)*(c/(c+d))^(1/2))*(g*sin(f*x+e)) 
^(1/2)*tan(f*x+e)/(a+b)/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx \] Input:

Integrate[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f* 
x]]),x]
 

Output:

Integrate[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f* 
x]]), x]
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3042, 3416}

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.

\(\displaystyle \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3416

\(\displaystyle \frac {2 \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {c \csc (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}\)

Input:

Int[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x 
]
 

Output:

(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2* 
a)/(a + b), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g* 
Sin[e + f*x]]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3416

Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. 
)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[2*Sqrt[ 
-Cot[e + f*x]^2]*(Sqrt[g*Sin[e + f*x]]/(f*(c + d)*Cot[e + f*x]*Sqrt[a + b*S 
in[e + f*x]]))*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[2*(c/(c + d)), 
 ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*(a/(a + b))], x] /; FreeQ[{a, b, 
 c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - 
d^2, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2581\) vs. \(2(107)=214\).

Time = 2.13 (sec) , antiderivative size = 2582, normalized size of antiderivative = 22.65

method	result	size
default	\(\text {Expression too large to display}\)	\(2582\)

Input:

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

Output:

1/f*a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c)/(a*(-c^2+d^2)^(1/2)+ 
c*(-a^2+b^2)^(1/2)+a*d-b*c)/(-a^2+b^2)^(1/2)*(EllipticPi((1/(d+(-c^2+d^2)^ 
(1/2))*(c*csc(f*x+e)-cot(f*x+e)*c+(-c^2+d^2)^(1/2)+d))^(1/2),(d+(-c^2+d^2) 
^(1/2))*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((d+ 
(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*a*c*d+EllipticPi((1/(d+(-c^2+d^ 
2)^(1/2))*(c*csc(f*x+e)-cot(f*x+e)*c+(-c^2+d^2)^(1/2)+d))^(1/2),(d+(-c^2+d 
^2)^(1/2))*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*( 
(d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*a*c*(-c^2+d^2)^(1/2)+Ellipti 
cPi((1/(d+(-c^2+d^2)^(1/2))*(c*csc(f*x+e)-cot(f*x+e)*c+(-c^2+d^2)^(1/2)+d) 
)^(1/2),(d+(-c^2+d^2)^(1/2))*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d- 
b*c),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*c^2-2*El 
lipticPi((1/(d+(-c^2+d^2)^(1/2))*(c*csc(f*x+e)-cot(f*x+e)*c+(-c^2+d^2)^(1/ 
2)+d))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2) 
+a*d-b*c),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*d^2 
-2*EllipticPi((1/(d+(-c^2+d^2)^(1/2))*(c*csc(f*x+e)-cot(f*x+e)*c+(-c^2+d^2 
)^(1/2)+d))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^ 
(1/2)+a*d-b*c),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))* 
b*d*(-c^2+d^2)^(1/2)-EllipticPi((1/(d+(-c^2+d^2)^(1/2))*(c*csc(f*x+e)-cot( 
f*x+e)*c+(-c^2+d^2)^(1/2)+d))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(a*(-c^2+d^2)^( 
1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^...
 

Fricas [F(-1)]

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

integrate((g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, 
algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(sqrt(g*sin(e + f*x))/((a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x 
))), x)
 

Maxima [F]

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

integrate((g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, 
algorithm="maxima")
 

Output:

integrate(sqrt(g*sin(f*x + e))/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + 
 c)), x)
 

Giac [F]

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

integrate((g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, 
algorithm="giac")
 

Output:

integrate(sqrt(g*sin(f*x + e))/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + 
 c)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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