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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2895} \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d} \]

[In]

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

[Out]

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

Rule 2895

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]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rubi steps \begin{align*} \text {integral}& = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.80 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\frac {8 a \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {a+b \sin (c+d x)}{a (-1+\sin (c+d x))}}\right ),\frac {2 a}{a-b}\right ) \sec (c+d x) \sqrt {-\frac {(a+b) \sin (c+d x) (a+b \sin (c+d x))}{a^2 (-1+\sin (c+d x))^2}} \sin ^4\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{(a+b) d \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(8*a*Sqrt[-(((a + b)*Cot[(2*c - Pi + 2*d*x)/4]^2)/(a - b))]*EllipticF[ArcSin[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]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(101)=202\).

Time = 4.03 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.42

method	result	size
default	\(-\frac {\sqrt {\frac {-\cot \left (d x +c \right ) a +\csc \left (d x +c \right ) a +\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {\sqrt {-a^{2}+b^{2}}+\cot \left (d x +c \right ) a -b -\csc \left (d x +c \right ) a}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {a \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, F\left (\sqrt {\frac {-\cot \left (d x +c \right ) a +\csc \left (d x +c \right ) a +\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \left (\sin ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}\, \left (b +\sqrt {-a^{2}+b^{2}}\right )}{d \sqrt {a +b \sin \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) a}\)	\(264\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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