\(\int \sin ^p(x) \, dx\) [86]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [F]
   Fricas [F]
   Sympy [F]
   Maxima [F]
   Giac [F]
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 4, antiderivative size = 44 \[ \int \sin ^p(x) \, dx=\frac {\cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2(x)\right ) \sin ^{1+p}(x)}{(1+p) \sqrt {\cos ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2722} \[ \int \sin ^p(x) \, dx=\frac {\cos (x) \sin ^{p+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {p+1}{2},\frac {p+3}{2},\sin ^2(x)\right )}{(p+1) \sqrt {\cos ^2(x)}} \]

[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 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x) \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \sin ^p(x) \, dx=\frac {\sqrt {\cos ^2(x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2(x)\right ) \sec (x) \sin ^{1+p}(x)}{1+p} \]

[In]

Integrate[Sin[x]^p,x]

[Out]

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

Maple [F]

\[\int \left (\sin ^{p}\left (x \right )\right )d x\]

[In]

int(sin(x)^p,x)

[Out]

int(sin(x)^p,x)

Fricas [F]

\[ \int \sin ^p(x) \, dx=\int { \sin \left (x\right )^{p} \,d x } \]

[In]

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

[Out]

integral(sin(x)^p, x)

Sympy [F]

\[ \int \sin ^p(x) \, dx=\int \sin ^{p}{\left (x \right )}\, dx \]

[In]

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

[Out]

Integral(sin(x)**p, x)

Maxima [F]

\[ \int \sin ^p(x) \, dx=\int { \sin \left (x\right )^{p} \,d x } \]

[In]

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

[Out]

integrate(sin(x)^p, x)

Giac [F]

\[ \int \sin ^p(x) \, dx=\int { \sin \left (x\right )^{p} \,d x } \]

[In]

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

[Out]

integrate(sin(x)^p, x)

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \sin ^p(x) \, dx=-\frac {\cos \left (x\right )\,{\sin \left (x\right )}^{p+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {1}{2}-\frac {p}{2};\ \frac {3}{2};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{\frac {p}{2}+\frac {1}{2}}} \]

[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)