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

Optimal. Leaf size=153 \[ \frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b f \sqrt{c+d \sin (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.328846, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2803, 2663, 2661, 2807, 2805} \[ \frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b f \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*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)])/(b*(a + b)*f*Sqrt[c + d*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 \frac{\sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx &=\frac{d \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{b}+\frac{(b c-a d) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{b}\\ &=\frac{\left (d \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{b \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}{b \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 d F\left (\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}}}{b 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}}}{b (a+b) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.8193, size = 114, normalized size = 0.75 \[ -\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d (a+b) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+(b c-a d) \Pi \left (\frac{2 b}{a+b};\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{b f (a+b) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 1.151, size = 181, normalized size = 1.2 \begin{align*} 2\,{\frac{c-d}{b\cos \left ( fx+e \right ) \sqrt{c+d\sin \left ( fx+e \right ) }f} \left ({\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) -{\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 ) \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)/(a+b*sin(f*x+e)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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