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

Optimal. Leaf size=44 \[ \frac {\cos (x) \sin ^{p+1}(x) \, _2F_1\left (\frac {1}{2},\frac {p+1}{2};\frac {p+3}{2};\sin ^2(x)\right )}{(p+1) \sqrt {\cos ^2(x)}} \]

[Out]

cos(x)*hypergeom([1/2, 1/2+1/2*p],[3/2+1/2*p],sin(x)^2)*sin(x)^(1+p)/(1+p)/(cos(x)^2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.04, size = 44, normalized size = 1.00 \[ -\cos (x) \sin ^{p+1}(x) \sin ^2(x)^{\frac {1}{2} (-p-1)} \, _2F_1\left (\frac {1}{2},\frac {1-p}{2};\frac {3}{2};\cos ^2(x)\right ) \]

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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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \relax (x)^{p}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \relax (x)^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)^p, x)

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maple [F]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \sin ^{p}\relax (x )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \relax (x)^{p}\,{d x} \]

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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mupad [B]  time = 0.32, size = 35, normalized size = 0.80 \[ -\frac {\cos \relax (x)\,{\sin \relax (x)}^{p+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {1}{2}-\frac {p}{2};\ \frac {3}{2};\ {\cos \relax (x)}^2\right )}{{\left ({\sin \relax (x)}^2\right )}^{\frac {p}{2}+\frac {1}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^p,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{p}{\relax (x )}\, dx \]

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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