3.117 \(\int \sin ^n(e+f x) \sqrt{1+\sin (e+f x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2 \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{\sin (e+f x)+1}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.047783, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2776, 65} \[ -\frac{2 \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{\sin (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]],x]

[Out]

(-2*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[1 + 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{1+\sin (e+f x)} \, dx &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{2 \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{1+\sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 0.408344, size = 186, normalized size = 4.33 \[ \frac{2^{1-n} e^{i (e+f x)} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{n+1} \sqrt{\sin (e+f x)+1} \left (i (2 n-1) \, _2F_1\left (1,\frac{1}{4} (2 n+3);\frac{1}{4} (3-2 n);e^{2 i (e+f x)}\right )+(2 n+1) e^{i (e+f x)} \, _2F_1\left (1,\frac{1}{4} (2 n+5);\frac{1}{4} (5-2 n);e^{2 i (e+f x)}\right )\right )}{f (2 n-1) (2 n+1) \left (e^{i (e+f x)}+i\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]],x]

[Out]

(2^(1 - n)*E^(I*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hypergeom
etric2F1[1, (3 + 2*n)/4, (3 - 2*n)/4, E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(1 + 2*n)*Hypergeometric2F1[1, (5
+ 2*n)/4, (5 - 2*n)/4, E^((2*I)*(e + f*x))])*Sqrt[1 + Sin[e + f*x]])/((I + E^(I*(e + f*x)))*f*(-1 + 2*n)*(1 +
2*n))

________________________________________________________________________________________

Maple [F]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( fx+e \right ) \right ) ^{n}\sqrt{1+\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*(1+sin(f*x+e))^(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(f*x + e)^n*sqrt(sin(f*x + e) + 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin(f*x + e)^n*sqrt(sin(f*x + e) + 1), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin{\left (e + f x \right )} + 1} \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*(1+sin(f*x+e))**(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(f*x + e)^n*sqrt(sin(f*x + e) + 1), x)