\(\int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [44]

Optimal result

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

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

Mathematica [F]

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

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

Output:

Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3416

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

Input:

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

Output:

(2*Sqrt[-Cot[e + f*x]^2]*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)]*Sqrt[g* 
Sin[e + f*x]]*Tan[e + f*x])/((c + d)*f*Sqrt[a + b*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. \(2589\) vs. \(2(107)=214\).

Time = 2.16 (sec) , antiderivative size = 2590, normalized size of antiderivative = 22.72

method	result	size
default	\(\text {Expression too large to display}\)	\(2590\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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