3.356 \(\int \sin ^6(2 x) \, dx\)

Optimal. Leaf size=46 \[ \frac{5 x}{16}-\frac{1}{12} \sin ^5(2 x) \cos (2 x)-\frac{5}{48} \sin ^3(2 x) \cos (2 x)-\frac{5}{32} \sin (2 x) \cos (2 x) \]

[Out]

(5*x)/16 - (5*Cos[2*x]*Sin[2*x])/32 - (5*Cos[2*x]*Sin[2*x]^3)/48 - (Cos[2*x]*Sin[2*x]^5)/12

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Rubi [A]  time = 0.0197261, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2635, 8} \[ \frac{5 x}{16}-\frac{1}{12} \sin ^5(2 x) \cos (2 x)-\frac{5}{48} \sin ^3(2 x) \cos (2 x)-\frac{5}{32} \sin (2 x) \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[2*x]^6,x]

[Out]

(5*x)/16 - (5*Cos[2*x]*Sin[2*x])/32 - (5*Cos[2*x]*Sin[2*x]^3)/48 - (Cos[2*x]*Sin[2*x]^5)/12

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(2 x) \, dx &=-\frac{1}{12} \cos (2 x) \sin ^5(2 x)+\frac{5}{6} \int \sin ^4(2 x) \, dx\\ &=-\frac{5}{48} \cos (2 x) \sin ^3(2 x)-\frac{1}{12} \cos (2 x) \sin ^5(2 x)+\frac{5}{8} \int \sin ^2(2 x) \, dx\\ &=-\frac{5}{32} \cos (2 x) \sin (2 x)-\frac{5}{48} \cos (2 x) \sin ^3(2 x)-\frac{1}{12} \cos (2 x) \sin ^5(2 x)+\frac{5 \int 1 \, dx}{16}\\ &=\frac{5 x}{16}-\frac{5}{32} \cos (2 x) \sin (2 x)-\frac{5}{48} \cos (2 x) \sin ^3(2 x)-\frac{1}{12} \cos (2 x) \sin ^5(2 x)\\ \end{align*}

Mathematica [A]  time = 0.0064757, size = 30, normalized size = 0.65 \[ \frac{5 x}{16}-\frac{15}{128} \sin (4 x)+\frac{3}{128} \sin (8 x)-\frac{1}{384} \sin (12 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[2*x]^6,x]

[Out]

(5*x)/16 - (15*Sin[4*x])/128 + (3*Sin[8*x])/128 - Sin[12*x]/384

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Maple [A]  time = 0.007, size = 32, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( 2\,x \right ) }{12} \left ( \left ( \sin \left ( 2\,x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( 2\,x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( 2\,x \right ) }{8}} \right ) }+{\frac{5\,x}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)^6,x)

[Out]

-1/12*(sin(2*x)^5+5/4*sin(2*x)^3+15/8*sin(2*x))*cos(2*x)+5/16*x

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Maxima [A]  time = 0.949246, size = 32, normalized size = 0.7 \begin{align*} \frac{1}{96} \, \sin \left (4 \, x\right )^{3} + \frac{5}{16} \, x + \frac{3}{128} \, \sin \left (8 \, x\right ) - \frac{1}{8} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)^6,x, algorithm="maxima")

[Out]

1/96*sin(4*x)^3 + 5/16*x + 3/128*sin(8*x) - 1/8*sin(4*x)

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Fricas [A]  time = 1.9265, size = 95, normalized size = 2.07 \begin{align*} -\frac{1}{96} \,{\left (8 \, \cos \left (2 \, x\right )^{5} - 26 \, \cos \left (2 \, x\right )^{3} + 33 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, x\right ) + \frac{5}{16} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)^6,x, algorithm="fricas")

[Out]

-1/96*(8*cos(2*x)^5 - 26*cos(2*x)^3 + 33*cos(2*x))*sin(2*x) + 5/16*x

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Sympy [A]  time = 0.059177, size = 46, normalized size = 1. \begin{align*} \frac{5 x}{16} - \frac{\sin ^{5}{\left (2 x \right )} \cos{\left (2 x \right )}}{12} - \frac{5 \sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{48} - \frac{5 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)**6,x)

[Out]

5*x/16 - sin(2*x)**5*cos(2*x)/12 - 5*sin(2*x)**3*cos(2*x)/48 - 5*sin(2*x)*cos(2*x)/32

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Giac [A]  time = 1.05715, size = 30, normalized size = 0.65 \begin{align*} \frac{5}{16} \, x - \frac{1}{384} \, \sin \left (12 \, x\right ) + \frac{3}{128} \, \sin \left (8 \, x\right ) - \frac{15}{128} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)^6,x, algorithm="giac")

[Out]

5/16*x - 1/384*sin(12*x) + 3/128*sin(8*x) - 15/128*sin(4*x)