\(\int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx\) [46]

Optimal result

Integrand size = 39, antiderivative size = 254 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 \sqrt {c+d} \sqrt {g} \sqrt {\frac {c (1-\csc (e+f x))}{c+d}} \sqrt {\frac {c (1+\csc (e+f x))}{c-d}} \operatorname {EllipticPi}\left (\frac {c+d}{d},\arcsin \left (\frac {\sqrt {g} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {g \sin (e+f x)}}\right ),-\frac {c+d}{c-d}\right ) \tan (e+f x)}{b f}+\frac {2 (b c-a d) \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {d+c \csc (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{b (a+b) f \sqrt {c+d \sin (e+f x)}} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3042, 3408, 3042, 3288, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3408

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3288

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

\(\Big \downarrow \) 3416

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

Input:

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

Output:

(2*Sqrt[c + d]*Sqrt[g]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*Sqrt[(c*(1 + C 
sc[e + f*x]))/(c - d)]*EllipticPi[(c + d)/d, ArcSin[(Sqrt[g]*Sqrt[c + d*Si 
n[e + f*x]])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]])], -((c + d)/(c - d))]*Tan[ 
e + f*x])/(b*f) + (2*(b*c - a*d)*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]]/Sq 
rt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(b*(a + b)*f*Sqr 
t[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 3288

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c 
*((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti 
cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + 
 d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - 
 d^2, 0] && PosQ[(c + d)/b]
 

rule 3408

Int[(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. 
)*(x_)]])/((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b/d   I 
nt[Sqrt[g*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] - Simp[(b*c - a*d) 
/d   Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x] 
)), x], 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]
 

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 [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.12 (sec) , antiderivative size = 5439, normalized size of antiderivative = 21.41

method	result	size
default	\(\text {Expression too large to display}\)	\(5439\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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