3.1.31 \(\int \sin ^5(x) \, dx\) [31]

Optimal. Leaf size=21 \[ -\cos (x)+\frac {2 \cos ^3(x)}{3}-\frac {\cos ^5(x)}{5} \]

[Out]

-cos(x)+2/3*cos(x)^3-1/5*cos(x)^5

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \begin {gather*} -\frac {1}{5} \cos ^5(x)+\frac {2 \cos ^3(x)}{3}-\cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^5,x]

[Out]

-Cos[x] + (2*Cos[x]^3)/3 - Cos[x]^5/5

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \sin ^5(x) \, dx &=-\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (x)\right )\\ &=-\cos (x)+\frac {2 \cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.10 \begin {gather*} -\frac {5 \cos (x)}{8}+\frac {5}{48} \cos (3 x)-\frac {1}{80} \cos (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^5,x]

[Out]

(-5*Cos[x])/8 + (5*Cos[3*x])/48 - Cos[5*x]/80

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Maple [A]
time = 0.10, size = 17, normalized size = 0.81

method	result	size
default	\(-\frac {\left (\frac {8}{3}+\sin ^{4}\left (x \right )+\frac {4 \left (\sin ^{2}\left (x \right )\right )}{3}\right ) \cos \left (x \right )}{5}\)	\(17\)
risch	\(-\frac {5 \cos \left (x \right )}{8}-\frac {\cos \left (5 x \right )}{80}+\frac {5 \cos \left (3 x \right )}{48}\)	\(18\)
norman	\(\frac {\frac {16 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{3}+\frac {16 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{15}+\frac {32 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5,x,method=_RETURNVERBOSE)

[Out]

-1/5*(8/3+sin(x)^4+4/3*sin(x)^2)*cos(x)

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Maxima [A]
time = 4.23, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{5} \, \cos \left (x\right )^{5} + \frac {2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^5,x, algorithm="maxima")

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Fricas [A]
time = 1.39, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{5} \, \cos \left (x\right )^{5} + \frac {2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^5,x, algorithm="fricas")

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Sympy [A]
time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} - \frac {\cos ^{5}{\left (x \right )}}{5} + \frac {2 \cos ^{3}{\left (x \right )}}{3} - \cos {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**5,x)

[Out]

-cos(x)**5/5 + 2*cos(x)**3/3 - cos(x)

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Giac [A]
time = 0.47, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{5} \, \cos \left (x\right )^{5} + \frac {2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^5,x, algorithm="giac")

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Mupad [B]
time = 0.04, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\cos \left (x\right )}^5}{5}+\frac {2\,{\cos \left (x\right )}^3}{3}-\cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5,x)

[Out]

(2*cos(x)^3)/3 - cos(x) - cos(x)^5/5

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