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

Optimal. Leaf size=60 \[ -\frac {2^{2 n+\frac {1}{2}} \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (\cos (c+d x)+1)\right )}{d \sqrt {1-\cos (c+d x)}} \]

[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)

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Rubi [A]  time = 0.02, antiderivative size = 60, 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} \[ -\frac {2^{2 n+\frac {1}{2}} \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (\cos (c+d x)+1)\right )}{d \sqrt {1-\cos (c+d x)}} \]

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*}

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Mathematica [A]  time = 0.07, size = 74, normalized size = 1.23 \[ \frac {\sqrt {2} \sqrt {\cos (c+d x)+1} \tan \left (\frac {1}{2} (c+d x)\right ) (2-2 \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},n+\frac {1}{2};n+\frac {3}{2};\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{2 d n+d} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 2*Cos[c + d*x])^n,x]

[Out]

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

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fricas [F]  time = 0.66, 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)

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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)

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maple [F]  time = 0.27, 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)

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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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2 - 2*cos(c + d*x))^n,x)

[Out]

int((4*sin(c/2 + (d*x)/2)^2)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((2 - 2*cos(c + d*x))**n, x)

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