\(\int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx\) [33]

Optimal result

Integrand size = 39, antiderivative size = 252 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {g E\left (\arcsin \left (\frac {\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac {a-b}{a+b}\right ) \sqrt {\frac {\sin (e+f x)}{1+\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}{(a-b) c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}-\frac {2 \sqrt {a+b} \sqrt {g} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{(a-b) c f} \] Output:

g*EllipticE(cos(f*x+e)/(1+sin(f*x+e)),(-(a-b)/(a+b))^(1/2))*(sin(f*x+e)/(1 
+sin(f*x+e)))^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a-b)/c/f/(g*sin(f*x+e))^(1/2)/ 
((a+b*sin(f*x+e))/(a+b)/(1+sin(f*x+e)))^(1/2)-2*(a+b)^(1/2)*g^(1/2)*(a*(1- 
csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*EllipticF(g^(1/2)* 
(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(g*sin(f*x+e))^(1/2),(-(a+b)/(a-b))^(1/ 
2))*tan(f*x+e)/(a-b)/c/f
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5708\) vs. \(2(252)=504\).

Time = 33.87 (sec) , antiderivative size = 5708, normalized size of antiderivative = 22.65 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 252, 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, 3415, 3042, 3295, 3411}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3415

\(\displaystyle \frac {a g \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx}{c (a-b)}-\frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx}{a-b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a g \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx}{c (a-b)}-\frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (\sin (e+f x) c+c)}dx}{a-b}\)

\(\Big \downarrow \) 3295

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

\(\Big \downarrow \) 3411

\(\displaystyle \frac {g \sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\arcsin \left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac {2 \sqrt {g} \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{c f (a-b)}\)

Input:

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

Output:

(g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]* 
Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c 
*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f* 
x]))]) - (2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[ 
(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e 
 + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + 
 f*x])/((a - b)*c*f)
 

Defintions of rubi rules used

rule 3042

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

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3411

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

rule 3415

Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. 
)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(-a)*(g 
/(b*c - a*d))   Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], 
x] + Simp[c*(g/(b*c - a*d))   Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + 
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N 
eQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1665\) vs. \(2(230)=460\).

Time = 4.10 (sec) , antiderivative size = 1666, normalized size of antiderivative = 6.61

method	result	size
default	\(\text {Expression too large to display}\)	\(1666\)

Input:

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

Output:

-1/2/c/f*(g*sin(f*x+e))^(1/2)*(a+b*sin(f*x+e))^(1/2)*(((-2*cos(f*x+e)-2)*s 
in(f*x+e)-2*(1+cos(f*x+e))^2)*(-a^2+b^2)^(1/2)*(-(a*cot(f*x+e)-(-a^2+b^2)^ 
(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2)*(1/(-a^2+b^2)^(1/2)*(a*c 
ot(f*x+e)+(-a^2+b^2)^(1/2)-b-a*csc(f*x+e)))^(1/2)*(1/(b+(-a^2+b^2)^(1/2))* 
a*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*EllipticE((-(a*cot(f*x+e)-(-a^2+b^2)^(1/ 
2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^ 
(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b+((2*cos(f*x+e)+2)*sin(f*x+e)+2*(1+cos(f* 
x+e))^2)*(-(a*cot(f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1 
/2)))^(1/2)*(1/(-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)*(1/(b+(-a^2+b^2)^(1/2))*a*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*Elli 
pticE((-(a*cot(f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2) 
))^(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*cos(f*x+e)-2)*sin(f*x+e)-2*(1+cos(f*x+e))^2)*(-(a*cot(f*x+e)-(-a^2+b^2) 
^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2)*(1/(-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)*(1/(b+(-a^2+b^2)^(1/2)) 
*a*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*EllipticE((-(a*cot(f*x+e)-(-a^2+b^2)^(1 
/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2) 
^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2+((1+cos(f*x+e))*sin(f*x+e)+(1+cos(f*x 
+e))^2)*(-a^2+b^2)^(1/2)*(-(a*cot(f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/ 
(b+(-a^2+b^2)^(1/2)))^(1/2)*(1/(-a^2+b^2)^(1/2)*(a*cot(f*x+e)+(-a^2+b^2...
 

Fricas [F]

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

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

Output:

integral(-sqrt(b*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/(b*c*cos(f*x + e)^ 
2 - (a + b)*c*sin(f*x + e) - (a + b)*c), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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