3.2.48 \(\int (1-\sin (c+d x))^n \, dx\) [148]

Optimal. Leaf size=57 \[ \frac {2^{\frac {1}{2}+n} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1+\sin (c+d x))\right )}{d \sqrt {1-\sin (c+d x)}} \]

[Out]

2^(1/2+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)

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Rubi [A]
time = 0.01, antiderivative size = 57, 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 = {2730} \begin {gather*} \frac {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)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Sin[c + d*x])^n,x]

[Out]

(2^(1/2 + 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 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]

Rubi steps

\begin {align*} \int (1-\sin (c+d x))^n \, dx &=\frac {2^{\frac {1}{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*}

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Mathematica [A]
time = 0.07, size = 90, normalized size = 1.58 \begin {gather*} \frac {\cos (c+d x) \cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )^{-\frac {1}{2}-n} \, _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-\pi +2 d x)\right )\right ) (1-\sin (c+d x))^n}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 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]*(1 - Sin[c + d*x])^n)/d

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (1-\sin \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-sin(d*x+c))^n,x)

[Out]

int((1-sin(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((1-sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((-sin(d*x + c) + 1)^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((1-sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((-sin(d*x + c) + 1)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (1 - \sin {\left (c + d x \right )}\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sin(d*x+c))**n,x)

[Out]

Integral((1 - sin(c + d*x))**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((1-sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((-sin(d*x + c) + 1)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (1-\sin \left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - sin(c + d*x))^n,x)

[Out]

int((1 - sin(c + d*x))^n, x)

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