3.351 \(\int \cos ^8(x) \sin ^8(x) \, dx\)

Optimal. Leaf size=90 \[ \frac{35 x}{32768}-\frac{1}{16} \sin ^7(x) \cos ^9(x)-\frac{1}{32} \sin ^5(x) \cos ^9(x)-\frac{5}{384} \sin ^3(x) \cos ^9(x)-\frac{1}{256} \sin (x) \cos ^9(x)+\frac{\sin (x) \cos ^7(x)}{2048}+\frac{7 \sin (x) \cos ^5(x)}{12288}+\frac{35 \sin (x) \cos ^3(x)}{49152}+\frac{35 \sin (x) \cos (x)}{32768} \]

[Out]

(35*x)/32768 + (35*Cos[x]*Sin[x])/32768 + (35*Cos[x]^3*Sin[x])/49152 + (7*Cos[x]^5*Sin[x])/12288 + (Cos[x]^7*S
in[x])/2048 - (Cos[x]^9*Sin[x])/256 - (5*Cos[x]^9*Sin[x]^3)/384 - (Cos[x]^9*Sin[x]^5)/32 - (Cos[x]^9*Sin[x]^7)
/16

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Rubi [A]  time = 0.136664, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{35 x}{32768}-\frac{1}{16} \sin ^7(x) \cos ^9(x)-\frac{1}{32} \sin ^5(x) \cos ^9(x)-\frac{5}{384} \sin ^3(x) \cos ^9(x)-\frac{1}{256} \sin (x) \cos ^9(x)+\frac{\sin (x) \cos ^7(x)}{2048}+\frac{7 \sin (x) \cos ^5(x)}{12288}+\frac{35 \sin (x) \cos ^3(x)}{49152}+\frac{35 \sin (x) \cos (x)}{32768} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^8*Sin[x]^8,x]

[Out]

(35*x)/32768 + (35*Cos[x]*Sin[x])/32768 + (35*Cos[x]^3*Sin[x])/49152 + (7*Cos[x]^5*Sin[x])/12288 + (Cos[x]^7*S
in[x])/2048 - (Cos[x]^9*Sin[x])/256 - (5*Cos[x]^9*Sin[x]^3)/384 - (Cos[x]^9*Sin[x]^5)/32 - (Cos[x]^9*Sin[x]^7)
/16

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

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 \cos ^8(x) \sin ^8(x) \, dx &=-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{7}{16} \int \cos ^8(x) \sin ^6(x) \, dx\\ &=-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{5}{32} \int \cos ^8(x) \sin ^4(x) \, dx\\ &=-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{5}{128} \int \cos ^8(x) \sin ^2(x) \, dx\\ &=-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{1}{256} \int \cos ^8(x) \, dx\\ &=\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{7 \int \cos ^6(x) \, dx}{2048}\\ &=\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{35 \int \cos ^4(x) \, dx}{12288}\\ &=\frac{35 \cos ^3(x) \sin (x)}{49152}+\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{35 \int \cos ^2(x) \, dx}{16384}\\ &=\frac{35 \cos (x) \sin (x)}{32768}+\frac{35 \cos ^3(x) \sin (x)}{49152}+\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{35 \int 1 \, dx}{32768}\\ &=\frac{35 x}{32768}+\frac{35 \cos (x) \sin (x)}{32768}+\frac{35 \cos ^3(x) \sin (x)}{49152}+\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)\\ \end{align*}

Mathematica [A]  time = 0.0148192, size = 38, normalized size = 0.42 \[ \frac{35 x}{32768}-\frac{7 \sin (4 x)}{16384}+\frac{7 \sin (8 x)}{65536}-\frac{\sin (12 x)}{49152}+\frac{\sin (16 x)}{524288} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^8*Sin[x]^8,x]

[Out]

(35*x)/32768 - (7*Sin[4*x])/16384 + (7*Sin[8*x])/65536 - Sin[12*x]/49152 + Sin[16*x]/524288

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Maple [A]  time = 0.033, size = 68, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{9} \left ( \sin \left ( x \right ) \right ) ^{7}}{16}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{9} \left ( \sin \left ( x \right ) \right ) ^{5}}{32}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{9} \left ( \sin \left ( x \right ) \right ) ^{3}}{384}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{9}\sin \left ( x \right ) }{256}}+{\frac{\sin \left ( x \right ) }{2048} \left ( \left ( \cos \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{32768}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^8*sin(x)^8,x)

[Out]

-1/16*cos(x)^9*sin(x)^7-1/32*cos(x)^9*sin(x)^5-5/384*cos(x)^9*sin(x)^3-1/256*cos(x)^9*sin(x)+1/2048*(cos(x)^7+
7/6*cos(x)^5+35/24*cos(x)^3+35/16*cos(x))*sin(x)+35/32768*x

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Maxima [A]  time = 0.944744, size = 41, normalized size = 0.46 \begin{align*} \frac{1}{12288} \, \sin \left (4 \, x\right )^{3} + \frac{35}{32768} \, x + \frac{1}{524288} \, \sin \left (16 \, x\right ) + \frac{7}{65536} \, \sin \left (8 \, x\right ) - \frac{1}{2048} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^8*sin(x)^8,x, algorithm="maxima")

[Out]

1/12288*sin(4*x)^3 + 35/32768*x + 1/524288*sin(16*x) + 7/65536*sin(8*x) - 1/2048*sin(4*x)

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Fricas [A]  time = 2.00414, size = 208, normalized size = 2.31 \begin{align*} \frac{1}{98304} \,{\left (6144 \, \cos \left (x\right )^{15} - 21504 \, \cos \left (x\right )^{13} + 25856 \, \cos \left (x\right )^{11} - 10880 \, \cos \left (x\right )^{9} + 48 \, \cos \left (x\right )^{7} + 56 \, \cos \left (x\right )^{5} + 70 \, \cos \left (x\right )^{3} + 105 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{35}{32768} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/98304*(6144*cos(x)^15 - 21504*cos(x)^13 + 25856*cos(x)^11 - 10880*cos(x)^9 + 48*cos(x)^7 + 56*cos(x)^5 + 70*
cos(x)^3 + 105*cos(x))*sin(x) + 35/32768*x

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Sympy [A]  time = 0.069368, size = 61, normalized size = 0.68 \begin{align*} \frac{35 x}{32768} - \frac{\sin ^{7}{\left (2 x \right )} \cos{\left (2 x \right )}}{4096} - \frac{7 \sin ^{5}{\left (2 x \right )} \cos{\left (2 x \right )}}{24576} - \frac{35 \sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{98304} - \frac{35 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{65536} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35*x/32768 - sin(2*x)**7*cos(2*x)/4096 - 7*sin(2*x)**5*cos(2*x)/24576 - 35*sin(2*x)**3*cos(2*x)/98304 - 35*sin
(2*x)*cos(2*x)/65536

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Giac [A]  time = 1.05709, size = 38, normalized size = 0.42 \begin{align*} \frac{35}{32768} \, x + \frac{1}{524288} \, \sin \left (16 \, x\right ) - \frac{1}{49152} \, \sin \left (12 \, x\right ) + \frac{7}{65536} \, \sin \left (8 \, x\right ) - \frac{7}{16384} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/32768*x + 1/524288*sin(16*x) - 1/49152*sin(12*x) + 7/65536*sin(8*x) - 7/16384*sin(4*x)