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

Optimal. Leaf size=172 \[ -\frac{2 \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 b f \sqrt{a+b \sin (e+f x)}}-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2 a \sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}} \]

[Out]

(-2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*f) + (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a +
b*Sin[e + f*x]])/(3*b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*(a^2 - b^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*
b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(3*b*f*Sqrt[a + b*Sin[e + f*x]])

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Rubi [A]  time = 0.169782, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 b f \sqrt{a+b \sin (e+f x)}}-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2 a \sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*f) + (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a +
b*Sin[e + f*x]])/(3*b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*(a^2 - b^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*
b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(3*b*f*Sqrt[a + b*Sin[e + f*x]])

Rule 2753

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

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 \sin (e+f x) \sqrt{a+b \sin (e+f x)} \, dx &=-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2}{3} \int \frac{\frac{b}{2}+\frac{1}{2} a \sin (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx\\ &=-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{a \int \sqrt{a+b \sin (e+f x)} \, dx}{3 b}-\frac{\left (a^2-b^2\right ) \int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx}{3 b}\\ &=-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{\left (a \sqrt{a+b \sin (e+f x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}} \, dx}{3 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \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}{3 b \sqrt{a+b \sin (e+f x)}}\\ &=-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2 a E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (e+f x)}}{3 b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) 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}}}{3 b f \sqrt{a+b \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 3.15589, size = 143, normalized size = 0.83 \[ -\frac{2 \left (-\left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )+b \cos (e+f x) (a+b \sin (e+f x))+a (a+b) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )\right )}{3 b f \sqrt{a+b \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*Cos[e + f*x]*(a + b*Sin[e + f*x]) + a*(a + b)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a +
b*Sin[e + f*x])/(a + b)] - (a^2 - b^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*
x])/(a + b)]))/(3*b*f*Sqrt[a + b*Sin[e + f*x]])

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Maple [B]  time = 0.918, size = 460, normalized size = 2.7 \begin{align*}{\frac{2}{3\,{b}^{2}\cos \left ( fx+e \right ) f} \left ( \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}}}{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{2}b-\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}}}{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){b}^{3}-\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}}}{\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{3}+\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}}}{\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{2}+ \left ( \sin \left ( fx+e \right ) \right ) ^{3}{b}^{3}+ \left ( \sin \left ( fx+e \right ) \right ) ^{2}a{b}^{2}-{b}^{3}\sin \left ( fx+e \right ) -a{b}^{2} \right ){\frac{1}{\sqrt{a+b\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((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)*EllipticF
(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*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)*EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^3
-((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)*EllipticE(((a
+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3+((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)*EllipticE(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^2+sin
(f*x+e)^3*b^3+sin(f*x+e)^2*a*b^2-b^3*sin(f*x+e)-a*b^2)/b^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} \sin \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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