3.86 \(\int \sin ^p(x) \, dx\)

Optimal. Leaf size=44 \[ \frac{\cos (x) \sin ^{p+1}(x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2(x)\right )}{(p+1) \sqrt{\cos ^2(x)}} \]

[Out]

(Cos[x]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[x]^2]*Sin[x]^(1 + p))/((1 + p)*Sqrt[Cos[x]^2])

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Rubi [A]  time = 0.0076611, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2643} \[ \frac{\cos (x) \sin ^{p+1}(x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2(x)\right )}{(p+1) \sqrt{\cos ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^p,x]

[Out]

(Cos[x]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[x]^2]*Sin[x]^(1 + p))/((1 + p)*Sqrt[Cos[x]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \sin ^p(x) \, dx &=\frac{\cos (x) \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};\sin ^2(x)\right ) \sin ^{1+p}(x)}{(1+p) \sqrt{\cos ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0395527, size = 44, normalized size = 1. \[ -\cos (x) \sin ^{p+1}(x) \sin ^2(x)^{\frac{1}{2} (-p-1)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-p}{2},\frac{3}{2},\cos ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^p,x]

[Out]

-(Cos[x]*Hypergeometric2F1[1/2, (1 - p)/2, 3/2, Cos[x]^2]*Sin[x]^(1 + p)*(Sin[x]^2)^((-1 - p)/2))

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Maple [F]  time = 0.385, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( x \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^p,x)

[Out]

int(sin(x)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (x\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^p,x, algorithm="maxima")

[Out]

integrate(sin(x)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (x\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^p,x, algorithm="fricas")

[Out]

integral(sin(x)^p, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{p}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**p,x)

[Out]

Integral(sin(x)**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (x\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^p,x, algorithm="giac")

[Out]

integrate(sin(x)^p, x)