Optimal. Leaf size=59 \[ \frac{2^{2 n+\frac{1}{2}} \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d \sqrt{\cos (c+d x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0184571, antiderivative size = 59, normalized size of antiderivative = 1., 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} \[ \frac{2^{2 n+\frac{1}{2}} \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d \sqrt{\cos (c+d x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2651
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*}
Mathematica [A] time = 0.093085, size = 77, normalized size = 1.31 \[ -\frac{2^{n+1} \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) (\cos (c+d x)+1)^n \, _2F_1\left (\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{2 d n+d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.368, size = 0, normalized size = 0. \begin{align*} \int \left ( 2+2\,\cos \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2^{n} \int \left (\cos{\left (c + d x \right )} + 1\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, \cos \left (d x + c\right ) + 2\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]