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

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

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Rubi [A]  time = 0.14, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.02, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^8(x) \sin ^8(x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.78, size = 55, normalized size = 0.61 \[ \frac {1}{98304} \, {\left (6144 \, \cos \relax (x)^{15} - 21504 \, \cos \relax (x)^{13} + 25856 \, \cos \relax (x)^{11} - 10880 \, \cos \relax (x)^{9} + 48 \, \cos \relax (x)^{7} + 56 \, \cos \relax (x)^{5} + 70 \, \cos \relax (x)^{3} + 105 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {35}{32768} \, x \]

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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giac [A]  time = 0.63, size = 28, normalized size = 0.31 \[ \frac {35}{32768} \, x + \frac {1}{524288} \, \sin \left (16 \, x\right ) - \frac {1}{49152} \, \sin \left (12 \, x\right ) + \frac {7}{65536} \, \sin \left (8 \, x\right ) - \frac {7}{16384} \, \sin \left (4 \, x\right ) \]

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)

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maple [A]  time = 0.32, size = 29, normalized size = 0.32

method	result	size
risch	\(\frac {35 x}{32768}+\frac {\sin \left (16 x \right )}{524288}-\frac {\sin \left (12 x \right )}{49152}+\frac {7 \sin \left (8 x \right )}{65536}-\frac {7 \sin \left (4 x \right )}{16384}\)	\(29\)
default	\(-\frac {\left (\cos ^{9}\relax (x )\right ) \left (\sin ^{7}\relax (x )\right )}{16}-\frac {\left (\cos ^{9}\relax (x )\right ) \left (\sin ^{5}\relax (x )\right )}{32}-\frac {5 \left (\cos ^{9}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{384}-\frac {\left (\cos ^{9}\relax (x )\right ) \sin \relax (x )}{256}+\frac {\left (\cos ^{7}\relax (x )+\frac {7 \left (\cos ^{5}\relax (x )\right )}{6}+\frac {35 \left (\cos ^{3}\relax (x )\right )}{24}+\frac {35 \cos \relax (x )}{16}\right ) \sin \relax (x )}{2048}+\frac {35 x}{32768}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^8*sin(x)^8,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.65, size = 30, normalized size = 0.33 \[ \frac {1}{12288} \, \sin \left (4 \, x\right )^{3} + \frac {35}{32768} \, x + \frac {1}{524288} \, \sin \left (16 \, x\right ) + \frac {7}{65536} \, \sin \left (8 \, x\right ) - \frac {1}{2048} \, \sin \left (4 \, x\right ) \]

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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mupad [B]  time = 0.04, size = 56, normalized size = 0.62 \[ \left (\frac {{\cos \relax (x)}^7}{16}+\frac {{\cos \relax (x)}^5}{32}+\frac {5\,{\cos \relax (x)}^3}{384}+\frac {\cos \relax (x)}{256}\right )\,{\sin \relax (x)}^9+\frac {35\,x}{32768}-\frac {7\,\sin \left (2\,x\right )}{8192}+\frac {7\,\sin \left (4\,x\right )}{32768}-\frac {\sin \left (6\,x\right )}{24576}+\frac {\sin \left (8\,x\right )}{262144} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35*x)/32768 - (7*sin(2*x))/8192 + (7*sin(4*x))/32768 - sin(6*x)/24576 + sin(8*x)/262144 + sin(x)^9*(cos(x)/25
6 + (5*cos(x)^3)/384 + cos(x)^5/32 + cos(x)^7/16)

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sympy [A]  time = 0.08, size = 61, normalized size = 0.68 \[ \frac {35 x}{32768} - \frac {\sin ^{7}{\left (2 x \right )} \cos {\left (2 x \right )}}{4096} - \frac {7 \sin ^{5}{\left (2 x \right )} \cos {\left (2 x \right )}}{24576} - \frac {35 \sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{98304} - \frac {35 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{65536} \]

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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