3.14 \(\int (a+a \cos (c+d x))^n \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0345538, antiderivative size = 73, normalized size of antiderivative = 1., 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 = {2652, 2651} \[ \frac{2^{n+\frac{1}{2}} \sin (c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{2}} (a \cos (c+d x)+a)^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d} \]

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 2652

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart
[n])/(1 + (b*Sin[c + d*x])/a)^FracPart[n], Int[(1 + (b*Sin[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

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 (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*}

Mathematica [A]  time = 0.0740444, size = 74, normalized size = 1.01 \[ -\frac{2 \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) (a (\cos (c+d x)+1))^n \, _2F_1\left (\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{2 d n+d} \]

Antiderivative was successfully verified.

[In]

Integrate[(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)

________________________________________________________________________________________

Maple [F]  time = 0.527, size = 0, normalized size = 0. \begin{align*} \int \left ( a+\cos \left ( dx+c \right ) a \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \cos \left (d x + c\right ) + a\right )}^{n}, x\right ) \end{align*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}

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)