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

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

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 8

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

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 \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*}

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

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sin[x]^8,x]

[Out]

Could not integrate

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fricas [A]  time = 0.97, size = 31, normalized size = 0.70 \[ \frac {1}{384} \, {\left (48 \, \cos \relax (x)^{7} - 200 \, \cos \relax (x)^{5} + 326 \, \cos \relax (x)^{3} - 279 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {35}{128} \, x \]

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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giac [A]  time = 0.61, size = 28, normalized size = 0.64 \[ \frac {35}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{96} \, \sin \left (6 \, x\right ) + \frac {7}{128} \, \sin \left (4 \, x\right ) - \frac {7}{32} \, \sin \left (2 \, x\right ) \]

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)

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maple [A]  time = 0.36, size = 29, normalized size = 0.66

method	result	size
risch	\(\frac {35 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (6 x \right )}{96}+\frac {7 \sin \left (4 x \right )}{128}-\frac {7 \sin \left (2 x \right )}{32}\)	\(29\)
default	\(-\frac {\left (\sin ^{7}\relax (x )+\frac {7 \left (\sin ^{5}\relax (x )\right )}{6}+\frac {35 \left (\sin ^{3}\relax (x )\right )}{24}+\frac {35 \sin \relax (x )}{16}\right ) \cos \relax (x )}{8}+\frac {35 x}{128}\)	\(30\)
norman	\(\frac {\frac {35 x}{128}-\frac {35 \tan \left (\frac {x}{2}\right )}{64}+\frac {1225 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{64}+\frac {245 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{16}+\frac {245 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{32}+\frac {245 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}-\frac {805 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{192}+\frac {35 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{16}+\frac {245 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{32}+\frac {35 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{16}+\frac {35 x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{128}+\frac {805 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{192}+\frac {35 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{64}-\frac {2681 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{192}-\frac {5053 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{192}+\frac {5053 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{192}+\frac {2681 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{192}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{8}}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 30, normalized size = 0.68 \[ \frac {1}{24} \, \sin \left (2 \, x\right )^{3} + \frac {35}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) + \frac {7}{128} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \]

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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mupad [B]  time = 0.03, size = 28, normalized size = 0.64 \[ \frac {35\,x}{128}-\frac {7\,\sin \left (2\,x\right )}{32}+\frac {7\,\sin \left (4\,x\right )}{128}-\frac {\sin \left (6\,x\right )}{96}+\frac {\sin \left (8\,x\right )}{1024} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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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sympy [A]  time = 0.07, size = 48, normalized size = 1.09 \[ \frac {35 x}{128} - \frac {\sin ^{7}{\relax (x )} \cos {\relax (x )}}{8} - \frac {7 \sin ^{5}{\relax (x )} \cos {\relax (x )}}{48} - \frac {35 \sin ^{3}{\relax (x )} \cos {\relax (x )}}{192} - \frac {35 \sin {\relax (x )} \cos {\relax (x )}}{128} \]

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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