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]
________________________________________________________________________________________
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]
[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 = 0.07, 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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________