\(\int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx\) [212]

Optimal result

Integrand size = 25, antiderivative size = 109 \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (c+d x)}{a d} \] Output:

-2*(a+b)^(1/2)*(a*(1-csc(d*x+c))/(a+b))^(1/2)*(a*(1+csc(d*x+c))/(a-b))^(1/ 
2)*EllipticF((a+b*sin(d*x+c))^(1/2)/(a+b)^(1/2)/sin(d*x+c)^(1/2),(-(a+b)/( 
a-b))^(1/2))*tan(d*x+c)/a/d
 

Mathematica [A] (warning: unable to verify)

Time = 5.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\frac {8 a \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {a+b \sin (c+d x)}{a (-1+\sin (c+d x))}}\right ),\frac {2 a}{a-b}\right ) \sec (c+d x) \sqrt {-\frac {(a+b) \sin (c+d x) (a+b \sin (c+d x))}{a^2 (-1+\sin (c+d x))^2}} \sin ^4\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{(a+b) d \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \] Input:

Integrate[1/(Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]),x]
 

Output:

(8*a*Sqrt[-(((a + b)*Cot[(2*c - Pi + 2*d*x)/4]^2)/(a - b))]*EllipticF[ArcS 
in[Sqrt[-((a + b*Sin[c + d*x])/(a*(-1 + Sin[c + d*x])))]], (2*a)/(a - b)]* 
Sec[c + d*x]*Sqrt[-(((a + b)*Sin[c + d*x]*(a + b*Sin[c + d*x]))/(a^2*(-1 + 
 Sin[c + d*x])^2))]*Sin[(2*c - Pi + 2*d*x)/4]^4)/((a + b)*d*Sqrt[Sin[c + d 
*x]]*Sqrt[a + b*Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3042, 3295}

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.

\(\displaystyle \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}}dx\)

\(\Big \downarrow \) 3295

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

Input:

Int[1/(Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]),x]
 

Output:

(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + 
d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqr 
t[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d)
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(98)=196\).

Time = 1.89 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.44

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {\frac {-\cot \left (d x +c \right ) a +\sqrt {-a^{2}+b^{2}}+b +\csc \left (d x +c \right ) a}{b +\sqrt {-a^{2}+b^{2}}}}\, \sqrt {-\frac {\csc \left (d x +c \right ) a -\cot \left (d x +c \right ) a -\sqrt {-a^{2}+b^{2}}+b}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {-\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) a}{b +\sqrt {-a^{2}+b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-\cot \left (d x +c \right ) a +\sqrt {-a^{2}+b^{2}}+b +\csc \left (d x +c \right ) a}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sin \left (d x +c \right )^{\frac {3}{2}} \left (b +\sqrt {-a^{2}+b^{2}}\right )}{d \sqrt {a +b \sin \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) a}\)	\(266\)

Input:

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

Output:

-1/d*2^(1/2)/(a+b*sin(d*x+c))^(1/2)*((-cot(d*x+c)*a+(-a^2+b^2)^(1/2)+b+csc 
(d*x+c)*a)/(b+(-a^2+b^2)^(1/2)))^(1/2)*(-(csc(d*x+c)*a-cot(d*x+c)*a-(-a^2+ 
b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-(csc(d*x+c)-cot(d*x+c))/(b+(-a^2+b 
^2)^(1/2))*a)^(1/2)*EllipticF(((-cot(d*x+c)*a+(-a^2+b^2)^(1/2)+b+csc(d*x+c 
)*a)/(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))*sin(d*x+c)^(3/2)/(-1+cos(d*x+c))*(b+(-a^2+b^2)^(1/2))/a
 

Fricas [F]

\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \sin {\left (c + d x \right )}} \sqrt {\sin {\left (c + d x \right )}}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(a + b*sin(c + d*x))*sqrt(sin(c + d*x))), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(b*sin(d*x + c) + a)*sqrt(sin(d*x + c))), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(b*sin(d*x + c) + a)*sqrt(sin(d*x + c))), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {1}{\sqrt {\sin \left (c+d\,x\right )}\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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