3.333 \(\int \cos ^6(x) \, dx\)

Optimal. Leaf size=34 \[ \frac {5 x}{16}+\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{24} \sin (x) \cos ^3(x)+\frac {5}{16} \sin (x) \cos (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2635, 8} \[ \frac {5 x}{16}+\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{24} \sin (x) \cos ^3(x)+\frac {5}{16} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^6,x]

[Out]

(5*x)/16 + (5*Cos[x]*Sin[x])/16 + (5*Cos[x]^3*Sin[x])/24 + (Cos[x]^5*Sin[x])/6

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 \cos ^6(x) \, dx &=\frac {1}{6} \cos ^5(x) \sin (x)+\frac {5}{6} \int \cos ^4(x) \, dx\\ &=\frac {5}{24} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)+\frac {5}{8} \int \cos ^2(x) \, dx\\ &=\frac {5}{16} \cos (x) \sin (x)+\frac {5}{24} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)+\frac {5 \int 1 \, dx}{16}\\ &=\frac {5 x}{16}+\frac {5}{16} \cos (x) \sin (x)+\frac {5}{24} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 0.88 \[ \frac {5 x}{16}+\frac {15}{64} \sin (2 x)+\frac {3}{64} \sin (4 x)+\frac {1}{192} \sin (6 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^6,x]

[Out]

(5*x)/16 + (15*Sin[2*x])/64 + (3*Sin[4*x])/64 + Sin[6*x]/192

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^6(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^6,x]

[Out]

Could not integrate

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fricas [A]  time = 0.67, size = 25, normalized size = 0.74 \[ \frac {1}{48} \, {\left (8 \, \cos \relax (x)^{5} + 10 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {5}{16} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6,x, algorithm="fricas")

[Out]

1/48*(8*cos(x)^5 + 10*cos(x)^3 + 15*cos(x))*sin(x) + 5/16*x

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giac [A]  time = 0.58, size = 22, normalized size = 0.65 \[ \frac {5}{16} \, x + \frac {1}{192} \, \sin \left (6 \, x\right ) + \frac {3}{64} \, \sin \left (4 \, x\right ) + \frac {15}{64} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6,x, algorithm="giac")

[Out]

5/16*x + 1/192*sin(6*x) + 3/64*sin(4*x) + 15/64*sin(2*x)

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maple [A]  time = 0.32, size = 23, normalized size = 0.68

method	result	size
risch	\(\frac {5 x}{16}+\frac {\sin \left (6 x \right )}{192}+\frac {3 \sin \left (4 x \right )}{64}+\frac {15 \sin \left (2 x \right )}{64}\)	\(23\)
default	\(\frac {\left (\cos ^{5}\relax (x )+\frac {5 \left (\cos ^{3}\relax (x )\right )}{4}+\frac {15 \cos \relax (x )}{8}\right ) \sin \relax (x )}{6}+\frac {5 x}{16}\)	\(24\)
norman	\(\frac {\frac {5 x}{16}-\frac {5 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {15 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}-\frac {15 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {5 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{24}-\frac {11 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{8}+\frac {15 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {75 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {25 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{4}+\frac {75 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{16}+\frac {15 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{8}+\frac {5 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{16}+\frac {11 \tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6,x,method=_RETURNVERBOSE)

[Out]

5/16*x+1/192*sin(6*x)+3/64*sin(4*x)+15/64*sin(2*x)

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maxima [A]  time = 0.44, size = 24, normalized size = 0.71 \[ -\frac {1}{48} \, \sin \left (2 \, x\right )^{3} + \frac {5}{16} \, x + \frac {3}{64} \, \sin \left (4 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6,x, algorithm="maxima")

[Out]

-1/48*sin(2*x)^3 + 5/16*x + 3/64*sin(4*x) + 1/4*sin(2*x)

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mupad [B]  time = 0.04, size = 22, normalized size = 0.65 \[ \frac {5\,x}{16}+\frac {15\,\sin \left (2\,x\right )}{64}+\frac {3\,\sin \left (4\,x\right )}{64}+\frac {\sin \left (6\,x\right )}{192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6,x)

[Out]

(5*x)/16 + (15*sin(2*x))/64 + (3*sin(4*x))/64 + sin(6*x)/192

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sympy [A]  time = 0.07, size = 36, normalized size = 1.06 \[ \frac {5 x}{16} + \frac {\sin {\relax (x )} \cos ^{5}{\relax (x )}}{6} + \frac {5 \sin {\relax (x )} \cos ^{3}{\relax (x )}}{24} + \frac {5 \sin {\relax (x )} \cos {\relax (x )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**6,x)

[Out]

5*x/16 + sin(x)*cos(x)**5/6 + 5*sin(x)*cos(x)**3/24 + 5*sin(x)*cos(x)/16

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