### 3.14 $$\int (a+a \sin (c+d x))^n \, dx$$

Optimal. Leaf size=74 $-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d}$

[Out]

-((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/
2 - n)*(a + a*Sin[c + d*x])^n)/d)

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Rubi [A]  time = 0.0382396, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {2652, 2651} $-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + a*Sin[c + d*x])^n,x]

[Out]

-((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/
2 - n)*(a + a*Sin[c + d*x])^n)/d)

Rule 2652

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart
[n])/(1 + (b*Sin[c + d*x])/a)^FracPart[n], Int[(1 + (b*Sin[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 2651

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(2^(n + 1/2)*a^(n - 1/2)*b*Cos[c + d*x]*Hy
pergeometric2F1[1/2, 1/2 - n, 3/2, (1*(1 - (b*Sin[c + d*x])/a))/2])/(d*Sqrt[a + b*Sin[c + d*x]]), x] /; FreeQ[
{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.187684, size = 90, normalized size = 1.22 $\frac{\sqrt{2} \cos (c+d x) (a (\sin (c+d x)+1))^n \, _2F_1\left (\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left (\frac{1}{4} (2 c+2 d x-\pi )\right )\right )}{(2 d n+d) \sqrt{1-\sin (c+d x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + a*Sin[c + d*x])^n,x]

[Out]

(Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]
*(a*(1 + Sin[c + d*x]))^n)/((d + 2*d*n)*Sqrt[1 - Sin[c + d*x]])

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Maple [F]  time = 0.31, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^n,x)

[Out]

int((a+a*sin(d*x+c))^n,x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^n, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((a*sin(d*x + c) + a)^n, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**n,x)

[Out]

Integral((a*sin(c + d*x) + a)**n, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^n, x)