3.40 \(\int \frac{\csc (e+f x) \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx\)

Optimal. Leaf size=154 \[ \frac{2 c \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f (a+b) \sqrt{c+d \sin (e+f x)}} \]

[Out]

(2*c*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[
e + f*x]]) - (2*(b*c - a*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f
*x])/(c + d)])/(a*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])

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Rubi [A]  time = 0.486799, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2935, 2807, 2805} \[ \frac{2 c \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f (a+b) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[
e + f*x]]) - (2*(b*c - a*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f
*x])/(c + d)])/(a*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2935

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])
), x_Symbol] :> Dist[a/c, Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Dist[(b*c - a*d)/c, Int[1/(S
qrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] &&
NeQ[a^2 - b^2, 0]

Rule 2807

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

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \frac{\csc (e+f x) \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx &=\frac{c \int \frac{\csc (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{a}+\frac{(-b c+a d) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{a}\\ &=\frac{\left (c \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{\csc (e+f x)}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{a \sqrt{c+d \sin (e+f x)}}+\frac{\left ((-b c+a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{a \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 c \Pi \left (2;\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{a f \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{a (a+b) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 3.73272, size = 179, normalized size = 1.16 \[ \frac{2 i \sec (e+f x) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}} \left (\Pi \left (\frac{c+d}{c};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-\Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )}{a f \sqrt{-\frac{1}{c+d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*(EllipticPi[(c + d)/c, I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - Elli
pticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sec
[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)])/(a*Sqrt[-(c + d)^(-1
)]*f)

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Maple [A]  time = 1.569, size = 190, normalized size = 1.2 \begin{align*} 2\,{\frac{c-d}{\cos \left ( fx+e \right ) a\sqrt{c+d\sin \left ( fx+e \right ) }f} \left ({\it EllipticPi} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},-{\frac{b \left ( c-d \right ) }{da-cb}},\sqrt{{\frac{c-d}{c+d}}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},{\frac{c-d}{c}},\sqrt{{\frac{c-d}{c+d}}} \right ) \right ) \sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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