3.1.15 \(\int (a-a \cos (c+d x))^n \, dx\) [15]

Optimal. Leaf size=75 \[ -\frac {2^{\frac {1}{2}+n} (1-\cos (c+d x))^{-\frac {1}{2}-n} (a-a \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1+\cos (c+d x))\right ) \sin (c+d x)}{d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2731, 2730} \begin {gather*} -\frac {2^{n+\frac {1}{2}} \sin (c+d x) (1-\cos (c+d x))^{-n-\frac {1}{2}} (a-a \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (\cos (c+d x)+1)\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - a*Cos[c + d*x])^n,x]

[Out]

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

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]

Rule 2731

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[a^IntPart[n]*((a + b*Sin[c + d*x])^FracPart
[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n]), Int[(1 + (b/a)*Sin[c + d*x])^n, 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 \cos (c+d x))^n \, dx &=\left ((1-\cos (c+d x))^{-n} (a-a \cos (c+d x))^n\right ) \int (1-\cos (c+d x))^n \, dx\\ &=-\frac {2^{\frac {1}{2}+n} (1-\cos (c+d x))^{-\frac {1}{2}-n} (a-a \cos (c+d x))^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}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 75, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \sqrt {1+\cos (c+d x)} (a-a \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}+n;\frac {3}{2}+n;\sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{d+2 d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Cos[c + d*x])^n,x]

[Out]

(Sqrt[2]*Sqrt[1 + Cos[c + d*x]]*(a - a*Cos[c + d*x])^n*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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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (a -a \cos \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-a*cos(d*x + c) + a)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-a*cos(d*x + c) + a)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- a \cos {\left (c + d x \right )} + a\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-a*cos(c + d*x) + a)**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-a*cos(d*x + c) + a)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a-a\,\cos \left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*cos(c + d*x))^n,x)

[Out]

int((a - a*cos(c + d*x))^n, x)

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