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

Optimal. Leaf size=170 \[ -\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

[Out]

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

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Rubi [A]  time = 0.207021, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2769, 2752, 2663, 2661, 2655, 2653} \[ -\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2769

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

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*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 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 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[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 2653

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

Rubi steps

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

Mathematica [A]  time = 1.09891, size = 201, normalized size = 1.18 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (2 \sin \left (\frac{1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left ((c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+c+d \sin (e+f x)\right )\right )}{a f (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]) - (Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])*(c + d*Sin[e + f*x] - (c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*
x])/(c + d)] + (c + d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])))/(
a*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])

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Maple [A]  time = 1.27, size = 382, normalized size = 2.3 \begin{align*}{\frac{1}{da\cos \left ( fx+e \right ) f}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) d+c \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( \sqrt{{\frac{d\sin \left ( fx+e \right ) }{c-d}}+{\frac{c}{c-d}}}\sqrt{-{\frac{d\sin \left ( fx+e \right ) }{c+d}}+{\frac{d}{c+d}}}\sqrt{-{\frac{d\sin \left ( fx+e \right ) }{c-d}}-{\frac{d}{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{d\sin \left ( fx+e \right ) }{c-d}}+{\frac{c}{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){c}^{2}-\sqrt{{\frac{d\sin \left ( fx+e \right ) }{c-d}}+{\frac{c}{c-d}}}\sqrt{-{\frac{d\sin \left ( fx+e \right ) }{c+d}}+{\frac{d}{c+d}}}\sqrt{-{\frac{d\sin \left ( fx+e \right ) }{c-d}}-{\frac{d}{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{d\sin \left ( fx+e \right ) }{c-d}}+{\frac{c}{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){d}^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}{d}^{2}+\sin \left ( fx+e \right ) cd-\sin \left ( fx+e \right ){d}^{2}-cd+{d}^{2} \right ){\frac{1}{\sqrt{- \left ( c+d\sin \left ( fx+e \right ) \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) }}}{\frac{1}{\sqrt{c+d\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(f*x+e)^2*sin(f*x+e)*d+c*cos(f*x+e)^2)^(1/2)*((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/
(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^
(1/2))*c^2-(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-
d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^2-cos(f*x+e)^2*d^2+sin(f*x+e)*
c*d-sin(f*x+e)*d^2-c*d+d^2)/d/(-(c+d*sin(f*x+e))*(-1+sin(f*x+e))*(1+sin(f*x+e)))^(1/2)/a/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}}{a \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+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*sin(e + f*x))/(sin(e + f*x) + 1), x)/a

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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}}{a \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+a*sin(f*x+e)),x, algorithm="giac")

[Out]

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