3.334 \(\int \sin ^8(x) \, dx\)

Optimal. Leaf size=44 \[ \frac{35 x}{128}-\frac{1}{8} \sin ^7(x) \cos (x)-\frac{7}{48} \sin ^5(x) \cos (x)-\frac{35}{192} \sin ^3(x) \cos (x)-\frac{35}{128} \sin (x) \cos (x) \]

[Out]

(35*x)/128 - (35*Cos[x]*Sin[x])/128 - (35*Cos[x]*Sin[x]^3)/192 - (7*Cos[x]*Sin[x]^5)/48 - (Cos[x]*Sin[x]^7)/8

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Rubi [A]  time = 0.0214529, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{35 x}{128}-\frac{1}{8} \sin ^7(x) \cos (x)-\frac{7}{48} \sin ^5(x) \cos (x)-\frac{35}{192} \sin ^3(x) \cos (x)-\frac{35}{128} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^8,x]

[Out]

(35*x)/128 - (35*Cos[x]*Sin[x])/128 - (35*Cos[x]*Sin[x]^3)/192 - (7*Cos[x]*Sin[x]^5)/48 - (Cos[x]*Sin[x]^7)/8

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 ^8(x) \, dx &=-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{7}{8} \int \sin ^6(x) \, dx\\ &=-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{35}{48} \int \sin ^4(x) \, dx\\ &=-\frac{35}{192} \cos (x) \sin ^3(x)-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{35}{64} \int \sin ^2(x) \, dx\\ &=-\frac{35}{128} \cos (x) \sin (x)-\frac{35}{192} \cos (x) \sin ^3(x)-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{35 \int 1 \, dx}{128}\\ &=\frac{35 x}{128}-\frac{35}{128} \cos (x) \sin (x)-\frac{35}{192} \cos (x) \sin ^3(x)-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)\\ \end{align*}

Mathematica [A]  time = 0.0025816, size = 38, normalized size = 0.86 \[ \frac{35 x}{128}-\frac{7}{32} \sin (2 x)+\frac{7}{128} \sin (4 x)-\frac{1}{96} \sin (6 x)+\frac{\sin (8 x)}{1024} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^8,x]

[Out]

(35*x)/128 - (7*Sin[2*x])/32 + (7*Sin[4*x])/128 - Sin[6*x]/96 + Sin[8*x]/1024

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Maple [A]  time = 0.034, size = 30, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( x \right ) }{8} \left ( \left ( \sin \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{128}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^8,x)

[Out]

-1/8*(sin(x)^7+7/6*sin(x)^5+35/24*sin(x)^3+35/16*sin(x))*cos(x)+35/128*x

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Maxima [A]  time = 0.940387, size = 41, normalized size = 0.93 \begin{align*} \frac{1}{24} \, \sin \left (2 \, x\right )^{3} + \frac{35}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) + \frac{7}{128} \, \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)^8,x, algorithm="maxima")

[Out]

1/24*sin(2*x)^3 + 35/128*x + 1/1024*sin(8*x) + 7/128*sin(4*x) - 1/4*sin(2*x)

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Fricas [A]  time = 1.69731, size = 111, normalized size = 2.52 \begin{align*} \frac{1}{384} \,{\left (48 \, \cos \left (x\right )^{7} - 200 \, \cos \left (x\right )^{5} + 326 \, \cos \left (x\right )^{3} - 279 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{35}{128} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^8,x, algorithm="fricas")

[Out]

1/384*(48*cos(x)^7 - 200*cos(x)^5 + 326*cos(x)^3 - 279*cos(x))*sin(x) + 35/128*x

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Sympy [A]  time = 0.059871, size = 48, normalized size = 1.09 \begin{align*} \frac{35 x}{128} - \frac{\sin ^{7}{\left (x \right )} \cos{\left (x \right )}}{8} - \frac{7 \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{48} - \frac{35 \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{192} - \frac{35 \sin{\left (x \right )} \cos{\left (x \right )}}{128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**8,x)

[Out]

35*x/128 - sin(x)**7*cos(x)/8 - 7*sin(x)**5*cos(x)/48 - 35*sin(x)**3*cos(x)/192 - 35*sin(x)*cos(x)/128

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Giac [A]  time = 1.07199, size = 38, normalized size = 0.86 \begin{align*} \frac{35}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) - \frac{1}{96} \, \sin \left (6 \, x\right ) + \frac{7}{128} \, \sin \left (4 \, x\right ) - \frac{7}{32} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^8,x, algorithm="giac")

[Out]

35/128*x + 1/1024*sin(8*x) - 1/96*sin(6*x) + 7/128*sin(4*x) - 7/32*sin(2*x)