3.1.14 \(\int (a+a \cos (c+d x))^n \, dx\) [14]

Optimal. Leaf size=73 \[ \frac {2^{\frac {1}{2}+n} (1+\cos (c+d x))^{-\frac {1}{2}-n} (a+a \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d} \]

[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 = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2731, 2730} \begin {gather*} \frac {2^{n+\frac {1}{2}} \sin (c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a \cos (c+d x)+a)^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\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 - Co
s[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 = 0.05, size = 74, normalized size = 1.01 \begin {gather*} -\frac {2 (a (1+\cos (c+d x)))^n \cot \left (\frac {1}{2} (c+d x)\right ) \, _2F_1\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^n,x]

[Out]

(-2*(a*(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]*Sqr
t[Sin[(c + d*x)/2]^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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