3.16 \(\int (2+2 \cos (c+d x))^n \, dx\)

Optimal. Leaf size=59 \[ \frac{2^{2 n+\frac{1}{2}} \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d \sqrt{\cos (c+d x)+1}} \]

[Out]

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

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Rubi [A]  time = 0.0184571, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2651} \[ \frac{2^{2 n+\frac{1}{2}} \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d \sqrt{\cos (c+d x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 2*Cos[c + d*x])^n,x]

[Out]

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

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

Mathematica [A]  time = 0.093085, size = 77, normalized size = 1.31 \[ -\frac{2^{n+1} \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) (\cos (c+d x)+1)^n \, _2F_1\left (\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{2 d n+d} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*Cos[c + d*x])^n,x]

[Out]

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

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Maple [F]  time = 0.368, size = 0, normalized size = 0. \begin{align*} \int \left ( 2+2\,\cos \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+2*cos(d*x+c))^n,x)

[Out]

int((2+2*cos(d*x+c))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+2*cos(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((2*cos(d*x + c) + 2)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+2*cos(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((2*cos(d*x + c) + 2)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2^{n} \int \left (\cos{\left (c + d x \right )} + 1\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+2*cos(d*x+c))**n,x)

[Out]

2**n*Integral((cos(c + d*x) + 1)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+2*cos(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((2*cos(d*x + c) + 2)^n, x)