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

Optimal. Leaf size=109 \[ -\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d} \]

[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]  time = 0.07, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2816} \[ -\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d} \]

Antiderivative was successfully verified.

[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 2816

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

Rubi steps

\begin {align*} \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}} F\left (\sin ^{-1}\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]  time = 3.34, size = 172, normalized size = 1.58 \[ \frac {8 a \sin ^4\left (\frac {1}{4} (2 c+2 d x-\pi )\right ) \sec (c+d x) \sqrt {-\frac {(a+b) \sin (c+d x) (a+b \sin (c+d x))}{a^2 (\sin (c+d x)-1)^2}} \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 c+2 d x-\pi )\right )}{a-b}} F\left (\sin ^{-1}\left (\sqrt {-\frac {a+b \sin (c+d x)}{a (\sin (c+d x)-1)}}\right )|\frac {2 a}{a-b}\right )}{d (a+b) \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[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]])

________________________________________________________________________________________

fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}}{b \cos \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - b}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [B]  time = 0.30, size = 318, normalized size = 2.92 \[ -\frac {\sqrt {-\frac {-\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )-b \sin \left (d x +c \right )+a \cos \left (d x +c \right )-a}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (d x +c \right )}}\, \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )-b \sin \left (d x +c \right )+a \cos \left (d x +c \right )-a}{\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )}}\, \sqrt {\frac {a \left (\cos \left (d x +c \right )-1\right )}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {-\frac {-\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )-b \sin \left (d x +c \right )+a \cos \left (d x +c \right )-a}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (d x +c \right )}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \left (\sin ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}\, \left (b +\sqrt {-a^{2}+b^{2}}\right )}{d \sqrt {a +b \sin \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {\sin \left (c+d\,x\right )}\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \sin {\left (c + d x \right )}} \sqrt {\sin {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________