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*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)

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Rubi [A]  time = 0.0677167, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, 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 = 3.26843, 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]])

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Maple [B]  time = 0.207, size = 310, normalized size = 2.8 \begin{align*} -{\frac{\sqrt{2}}{da \left ( -1+\cos \left ( dx+c \right ) \right ) } \left ( b+\sqrt{-{a}^{2}+{b}^{2}} \right ) \sqrt{{\frac{1}{\sin \left ( dx+c \right ) } \left ( \sqrt{-{a}^{2}+{b}^{2}}\sin \left ( dx+c \right ) +b\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) a+a \right ) \left ( b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}}}\sqrt{{\frac{1}{\sin \left ( dx+c \right ) } \left ( \sqrt{-{a}^{2}+{b}^{2}}\sin \left ( dx+c \right ) -b\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) a-a \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}}}\sqrt{{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) a}{\sin \left ( dx+c \right ) } \left ( b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}}}{\it EllipticF} \left ( \sqrt{{\frac{1}{\sin \left ( dx+c \right ) } \left ( \sqrt{-{a}^{2}+{b}^{2}}\sin \left ( dx+c \right ) +b\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) a+a \right ) \left ( b+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+\sqrt{-{a}^{2}+{b}^{2}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{a+b\sin \left ( dx+c \right ) }}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sin \left (d x + c\right ) + a} \sqrt{\sin \left (d x + c\right )}}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \sin{\left (c + d x \right )}} \sqrt{\sin{\left (c + d x \right )}}}\, dx \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sin \left (d x + c\right ) + a} \sqrt{\sin \left (d x + c\right )}}\,{d x} \end{align*}

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)