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)*hypergeom([1/2, 1/2-n],[3/2],1/2-1/2*cos(d*x+c))*sin(d*x+c)/d/(1+cos(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 = 0.09, 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))

________________________________________________________________________________________

fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (2+2 \cos \left (d x +c \right )\right )^{n}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*cos(c + d*x) + 2)^n,x)

[Out]

int((4*cos(c/2 + (d*x)/2)^2)^n, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 2^{n} \int \left (\cos {\left (c + d x \right )} + 1\right )^{n}\, dx \]

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)

________________________________________________________________________________________