\(\int (2+2 \cos (c+d x))^n \, dx\) [16]

Optimal result

Integrand size = 12, antiderivative size = 59 \[ \int (2+2 \cos (c+d x))^n \, dx=\frac {2^{\frac {1}{2}+2 n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}} \]

[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] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2730} \[ \int (2+2 \cos (c+d x))^n \, dx=\frac {2^{2 n+\frac {1}{2}} \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{d \sqrt {\cos (c+d x)+1}} \]

[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 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2^{\frac {1}{2}+2 n} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int (2+2 \cos (c+d x))^n \, dx=-\frac {2^{1+n} (1+\cos (c+d x))^n \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}+n,\frac {3}{2}+n,\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{d+2 d n} \]

[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))

Maple [F]

\[\int \left (2+2 \cos \left (d x +c \right )\right )^{n}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (2+2 \cos (c+d x))^n \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n} \,d x } \]

[In]

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

[Out]

integral((2*cos(d*x + c) + 2)^n, x)

Sympy [F]

\[ \int (2+2 \cos (c+d x))^n \, dx=2^{n} \int \left (\cos {\left (c + d x \right )} + 1\right )^{n}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (2+2 \cos (c+d x))^n \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n} \,d x } \]

[In]

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

[Out]

integrate((2*cos(d*x + c) + 2)^n, x)

Giac [F]

\[ \int (2+2 \cos (c+d x))^n \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n} \,d x } \]

[In]

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

[Out]

integrate((2*cos(d*x + c) + 2)^n, x)

Mupad [F(-1)]

Timed out. \[ \int (2+2 \cos (c+d x))^n \, dx=\int {\left (4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}^n \,d x \]

[In]

int((2*cos(c + d*x) + 2)^n,x)

[Out]

int((4*cos(c/2 + (d*x)/2)^2)^n, x)