3.3.88 \(\int (\cos ^6(x)+3 \cos ^2(x) \sin ^2(x)+\sin ^6(x)) \, dx\) [288]

Optimal. Leaf size=1 \[ x \]

[Out]

x

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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
time = 0.04, antiderivative size = 50, normalized size of antiderivative = 50.00, number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2715, 8, 2648} \begin {gather*} x+\frac {1}{6} \sin (x) \cos ^5(x)-\frac {13}{24} \sin (x) \cos ^3(x)-\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{24} \sin ^3(x) \cos (x)+\frac {3}{8} \sin (x) \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]^6 + 3*Cos[x]^2*Sin[x]^2 + Sin[x]^6,x]

[Out]

x + (3*Cos[x]*Sin[x])/8 - (13*Cos[x]^3*Sin[x])/24 + (Cos[x]^5*Sin[x])/6 - (5*Cos[x]*Sin[x]^3)/24 - (Cos[x]*Sin
[x]^5)/6

Rule 8

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

Rule 2648

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 2715

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] && Integ
erQ[2*n]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=3 \int \cos ^2(x) \sin ^2(x) \, dx+\int \cos ^6(x) \, dx+\int \sin ^6(x) \, dx\\ &=-\frac {3}{4} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)-\frac {1}{6} \cos (x) \sin ^5(x)+\frac {3}{4} \int \cos ^2(x) \, dx+\frac {5}{6} \int \cos ^4(x) \, dx+\frac {5}{6} \int \sin ^4(x) \, dx\\ &=\frac {3}{8} \cos (x) \sin (x)-\frac {13}{24} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x)+\frac {3 \int 1 \, dx}{8}+\frac {5}{8} \int \cos ^2(x) \, dx+\frac {5}{8} \int \sin ^2(x) \, dx\\ &=\frac {3 x}{8}+\frac {3}{8} \cos (x) \sin (x)-\frac {13}{24} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x)+2 \frac {5 \int 1 \, dx}{16}\\ &=x+\frac {3}{8} \cos (x) \sin (x)-\frac {13}{24} \cos ^3(x) \sin (x)+\frac {1}{6} \cos ^5(x) \sin (x)-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^6 + 3*Cos[x]^2*Sin[x]^2 + Sin[x]^6,x]

[Out]

x

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.16, size = 55, normalized size = 55.00

method	result	size
risch	\(x\)	\(2\)
default	\(\frac {\left (\cos ^{5}\left (x \right )+\frac {5 \left (\cos ^{3}\left (x \right )\right )}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{6}+x -\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}-\frac {3 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}{4}+\frac {3 \cos \left (x \right ) \sin \left (x \right )}{8}\)	\(55\)
parts	\(\frac {\left (\cos ^{5}\left (x \right )+\frac {5 \left (\cos ^{3}\left (x \right )\right )}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{6}+x -\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}-\frac {3 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}{4}+\frac {3 \cos \left (x \right ) \sin \left (x \right )}{8}\)	\(55\)
norman	\(\frac {x +x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )+6 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+15 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+20 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+15 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+6 \left (\tan ^{10}\left (\frac {x}{2}\right )\right ) x}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^6+cos(x)^6+3*sin(x)^2*cos(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/6*(cos(x)^5+5/4*cos(x)^3+15/8*cos(x))*sin(x)+x-1/6*(sin(x)^5+5/4*sin(x)^3+15/8*sin(x))*cos(x)-3/4*cos(x)^3*s
in(x)+3/8*cos(x)*sin(x)

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Maxima [A]
time = 0.40, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6+cos(x)^6+3*sin(x)^2*cos(x)^2,x, algorithm="maxima")

[Out]

x

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Fricas [A]
time = 0.55, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6+cos(x)^6+3*sin(x)^2*cos(x)^2,x, algorithm="fricas")

[Out]

x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (0) = 0\).
time = 0.02, size = 58, normalized size = 58.00 \begin {gather*} x - \frac {\sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{6} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{24} + \frac {\sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{6} + \frac {5 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{24} - \frac {3 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**6+cos(x)**6+3*sin(x)**2*cos(x)**2,x)

[Out]

x - sin(x)**5*cos(x)/6 - 5*sin(x)**3*cos(x)/24 + sin(x)*cos(x)**5/6 + 5*sin(x)*cos(x)**3/24 - 3*sin(2*x)*cos(2
*x)/16

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Giac [A]
time = 0.45, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^6+cos(x)^6+3*sin(x)^2*cos(x)^2,x, algorithm="giac")

[Out]

x

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Mupad [B]
time = 0.15, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6 + sin(x)^6 + 3*cos(x)^2*sin(x)^2,x)

[Out]

x

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Chatgpt [F] Failed to verify
time = 1.00, size = 35, normalized size = 35.00 \begin {gather*} \frac {x}{8}-\frac {\cos \left (6 x \right )}{64}-\frac {3 \cos \left (2 x \right )}{32}+\frac {3 x \sin \left (2 x \right )}{16}-\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{32} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(sin(x)^6+cos(x)^6+3*sin(x)^2*cos(x)^2,x)

[Out]

1/8*x-1/64*cos(6*x)-3/32*cos(2*x)+3/16*x*sin(2*x)-1/16*sin(4*x)+1/32*sin(6*x)

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