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

Optimal. Leaf size=72 \[ \frac{\cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},n+1;n+2;\sin (e+f x)\right )}{d f (n+1) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.055852, antiderivative size = 72, 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, 64} \[ \frac{\cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},n+1;n+2;\sin (e+f x)\right )}{d f (n+1) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + n, 2 + n, Sin[e + f*x]]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[1
- Sin[e + f*x]]*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 64

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

Rubi steps

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

Mathematica [C]  time = 0.306138, size = 215, normalized size = 2.99 \[ \frac{(1-i) 2^{-n} e^{\frac{1}{2} i (e+f x)} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{n+1} \sqrt{\sin (e+f x)+1} \left (i (2 n-1) \, _2F_1\left (1,\frac{1}{4} (2 n+3);\frac{1}{4} (3-2 n);e^{2 i (e+f x)}\right )+(2 n+1) e^{i (e+f x)} \, _2F_1\left (1,\frac{1}{4} (2 n+5);\frac{1}{4} (5-2 n);e^{2 i (e+f x)}\right )\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f (2 n-1) (2 n+1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I)*E^((I/2)*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hyperge
ometric2F1[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))])*(d*Sin[e + f*x])^n*Sqrt[1 + Sin[e + f*x]])/(2^n*f*(-1 + 2*n)*(
1 + 2*n)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]^n)

________________________________________________________________________________________

Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int \left ( d\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((d*sin(f*x+e))^n*(1+sin(f*x+e))^(1/2),x)

[Out]

int((d*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 \left (d \sin \left (f x + e\right )\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((d*sin(f*x+e))^n*(1+sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sin \left (f x + e\right )\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((d*sin(f*x+e))^n*(1+sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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