3.17 \(\int (2-2 \sin (c+d x))^n \, dx\)

Optimal. Leaf size=59 \[ \frac {2^{2 n+\frac {1}{2}} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 59, 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} \[ \frac {2^{2 n+\frac {1}{2}} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - 2*Sin[c + d*x])^n,x]

[Out]

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

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 (2-2 \sin (c+d x))^n \, dx &=\frac {2^{\frac {1}{2}+2 n} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1+\sin (c+d x))\right )}{d \sqrt {1-\sin (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 90, normalized size = 1.53 \[ \frac {\cos (c+d x) (2-2 \sin (c+d x))^n \cos ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )^{-n-\frac {1}{2}} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{4} \cos ^2(c+d x) \csc ^2\left (\frac {1}{4} (2 c+2 d x-\pi )\right )\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 - 2*Sin[c + d*x])^n,x]

[Out]

(Cos[c + d*x]*(Cos[(2*c + Pi + 2*d*x)/4]^2)^(-1/2 - n)*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (Cos[c + d*x]^2*Cs
c[(2*c - Pi + 2*d*x)/4]^2)/4]*(2 - 2*Sin[c + d*x])^n)/d

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-2 \, \sin \left (d x + c\right ) + 2\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-2*sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((-2*sin(d*x + c) + 2)^n, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-2 \, \sin \left (d x + c\right ) + 2\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-2*sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((-2*sin(d*x + c) + 2)^n, x)

________________________________________________________________________________________

maple [F]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \left (2-2 \sin \left (d x +c \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-2*sin(d*x+c))^n,x)

[Out]

int((2-2*sin(d*x+c))^n,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-2 \, \sin \left (d x + c\right ) + 2\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-2*sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((-2*sin(d*x + c) + 2)^n, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (2-2\,\sin \left (c+d\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2 - 2*sin(c + d*x))^n,x)

[Out]

int((2 - 2*sin(c + d*x))^n, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2 - 2 \sin {\left (c + d x \right )}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-2*sin(d*x+c))**n,x)

[Out]

Integral((2 - 2*sin(c + d*x))**n, x)

________________________________________________________________________________________