3.205 \(\int \csc (e+f x) \sqrt{a+b \sin (e+f x)} \, dx\)

Optimal. Leaf size=128 \[ \frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.235212, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2803, 2663, 2661, 2807, 2805} \[ \frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]

[Out]

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

Rule 2803

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

Rule 2663

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

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 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 \csc (e+f x) \sqrt{a+b \sin (e+f x)} \, dx &=a \int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx+b \int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx\\ &=\frac{\left (a \sqrt{\frac{a+b \sin (e+f x)}{a+b}}\right ) \int \frac{\csc (e+f x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{\sqrt{a+b \sin (e+f x)}}+\frac{\left (b \sqrt{\frac{a+b \sin (e+f x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{\sqrt{a+b \sin (e+f x)}}\\ &=\frac{2 b F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}{f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \Pi \left (2;\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}{f \sqrt{a+b \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 15.4153, size = 89, normalized size = 0.7 \[ -\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \left (b F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )+a \Pi \left (2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )\right )}{f \sqrt{a+b \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(-2*(b*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)] + a*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]
)*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])

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Maple [A]  time = 0.832, size = 169, normalized size = 1.3 \begin{align*} 2\,{\frac{a-b}{\cos \left ( fx+e \right ) \sqrt{a+b\sin \left ( fx+e \right ) }f}\sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( fx+e \right ) \right ) b}{a-b}}} \left ({\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},{\frac{a-b}{a}},\sqrt{{\frac{a-b}{a+b}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*csc(f*x + e), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sin(e + f*x))*csc(e + f*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*csc(f*x + e), x)