3.121 \(\int \sin ^n(e+f x) \sqrt{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 a \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a \sin (e+f x)+a}} \]

[Out]

(-2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 0.0615039, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2776, 65} \[ -\frac{2 a \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])

Rule 2776

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

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \sin ^n(e+f x) \sqrt{a+a \sin (e+f x)} \, dx &=\frac{\left (a^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 4.08545, size = 264, normalized size = 5.74 \[ \frac{(1+i) e^{-\frac{1}{2} i f x} \sqrt{a (\sin (e+f x)+1)} \sin ^n(e+f x) \left (\sin ^2(e) e^{2 i f x}-i \sin (2 e) e^{2 i f x}+\cos ^2(e) \left (-e^{2 i f x}\right )+1\right )^{-n} \left ((2 n+1) e^{i f x} \left (\cos \left (\frac{e}{2}\right )+i \sin \left (\frac{e}{2}\right )\right ) \, _2F_1\left (\frac{1}{4} (1-2 n),-n;\frac{1}{4} (5-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right )+(2 n-1) \left (\sin \left (\frac{e}{2}\right )+i \cos \left (\frac{e}{2}\right )\right ) \, _2F_1\left (\frac{1}{4} (-2 n-1),-n;\frac{1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right )\right )}{f (2 n-1) (2 n+1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + I)*(E^(I*f*x)*(1 + 2*n)*Hypergeometric2F1[(1 - 2*n)/4, -n, (5 - 2*n)/4, E^((2*I)*f*x)*(Cos[e] + I*Sin[e]
)^2]*(Cos[e/2] + I*Sin[e/2]) + (-1 + 2*n)*Hypergeometric2F1[(-1 - 2*n)/4, -n, (3 - 2*n)/4, E^((2*I)*f*x)*(Cos[
e] + I*Sin[e])^2]*(I*Cos[e/2] + Sin[e/2]))*Sin[e + f*x]^n*Sqrt[a*(1 + Sin[e + f*x])])/(E^((I/2)*f*x)*f*(-1 + 2
*n)*(1 + 2*n)*(1 - E^((2*I)*f*x)*Cos[e]^2 + E^((2*I)*f*x)*Sin[e]^2 - I*E^((2*I)*f*x)*Sin[2*e])^n*(Cos[(e + f*x
)/2] + Sin[(e + f*x)/2]))

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Maple [F]  time = 0.138, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( fx+e \right ) \right ) ^{n}\sqrt{a+a\sin \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*sin(e + f*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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