3.69 \(\int \sin ^6(x) \, dx\)

Optimal. Leaf size=34 \[ \frac{5 x}{16}-\frac{1}{6} \sin ^5(x) \cos (x)-\frac{5}{24} \sin ^3(x) \cos (x)-\frac{5}{16} \sin (x) \cos (x) \]

[Out]

(5*x)/16 - (5*Cos[x]*Sin[x])/16 - (5*Cos[x]*Sin[x]^3)/24 - (Cos[x]*Sin[x]^5)/6

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Rubi [A]  time = 0.0167524, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{5 x}{16}-\frac{1}{6} \sin ^5(x) \cos (x)-\frac{5}{24} \sin ^3(x) \cos (x)-\frac{5}{16} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^6,x]

[Out]

(5*x)/16 - (5*Cos[x]*Sin[x])/16 - (5*Cos[x]*Sin[x]^3)/24 - (Cos[x]*Sin[x]^5)/6

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sin ^6(x) \, dx &=-\frac{1}{6} \cos (x) \sin ^5(x)+\frac{5}{6} \int \sin ^4(x) \, dx\\ &=-\frac{5}{24} \cos (x) \sin ^3(x)-\frac{1}{6} \cos (x) \sin ^5(x)+\frac{5}{8} \int \sin ^2(x) \, dx\\ &=-\frac{5}{16} \cos (x) \sin (x)-\frac{5}{24} \cos (x) \sin ^3(x)-\frac{1}{6} \cos (x) \sin ^5(x)+\frac{5 \int 1 \, dx}{16}\\ &=\frac{5 x}{16}-\frac{5}{16} \cos (x) \sin (x)-\frac{5}{24} \cos (x) \sin ^3(x)-\frac{1}{6} \cos (x) \sin ^5(x)\\ \end{align*}

Mathematica [A]  time = 0.0021813, size = 30, normalized size = 0.88 \[ \frac{5 x}{16}-\frac{15}{64} \sin (2 x)+\frac{3}{64} \sin (4 x)-\frac{1}{192} \sin (6 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^6,x]

[Out]

(5*x)/16 - (15*Sin[2*x])/64 + (3*Sin[4*x])/64 - Sin[6*x]/192

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Maple [A]  time = 0., size = 24, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( x \right ) }{6} \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) }+{\frac{5\,x}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^6,x)

[Out]

-1/6*(sin(x)^5+5/4*sin(x)^3+15/8*sin(x))*cos(x)+5/16*x

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Maxima [A]  time = 0.924352, size = 32, normalized size = 0.94 \begin{align*} \frac{1}{48} \, \sin \left (2 \, x\right )^{3} + \frac{5}{16} \, x + \frac{3}{64} \, \sin \left (4 \, x\right ) - \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6,x, algorithm="maxima")

[Out]

1/48*sin(2*x)^3 + 5/16*x + 3/64*sin(4*x) - 1/4*sin(2*x)

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Fricas [A]  time = 1.99774, size = 84, normalized size = 2.47 \begin{align*} -\frac{1}{48} \,{\left (8 \, \cos \left (x\right )^{5} - 26 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{5}{16} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6,x, algorithm="fricas")

[Out]

-1/48*(8*cos(x)^5 - 26*cos(x)^3 + 33*cos(x))*sin(x) + 5/16*x

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Sympy [A]  time = 0.061398, size = 36, normalized size = 1.06 \begin{align*} \frac{5 x}{16} - \frac{\sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{6} - \frac{5 \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{24} - \frac{5 \sin{\left (x \right )} \cos{\left (x \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**6,x)

[Out]

5*x/16 - sin(x)**5*cos(x)/6 - 5*sin(x)**3*cos(x)/24 - 5*sin(x)*cos(x)/16

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Giac [A]  time = 1.05327, size = 30, normalized size = 0.88 \begin{align*} \frac{5}{16} \, x - \frac{1}{192} \, \sin \left (6 \, x\right ) + \frac{3}{64} \, \sin \left (4 \, x\right ) - \frac{15}{64} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6,x, algorithm="giac")

[Out]

5/16*x - 1/192*sin(6*x) + 3/64*sin(4*x) - 15/64*sin(2*x)