3.1.53 \(\int x^5 \cos (x^3) \, dx\) [53]

Optimal. Leaf size=20 \[ \frac {\cos \left (x^3\right )}{3}+\frac {1}{3} x^3 \sin \left (x^3\right ) \]

[Out]

1/3*cos(x^3)+1/3*x^3*sin(x^3)

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3461, 3377, 2718} \begin {gather*} \frac {1}{3} x^3 \sin \left (x^3\right )+\frac {\cos \left (x^3\right )}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Cos[x^3],x]

[Out]

Cos[x^3]/3 + (x^3*Sin[x^3])/3

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps

\begin {align*} \int x^5 \cos \left (x^3\right ) \, dx &=\frac {1}{3} \text {Subst}\left (\int x \cos (x) \, dx,x,x^3\right )\\ &=\frac {1}{3} x^3 \sin \left (x^3\right )-\frac {1}{3} \text {Subst}\left (\int \sin (x) \, dx,x,x^3\right )\\ &=\frac {\cos \left (x^3\right )}{3}+\frac {1}{3} x^3 \sin \left (x^3\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {\cos \left (x^3\right )}{3}+\frac {1}{3} x^3 \sin \left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Cos[x^3],x]

[Out]

Cos[x^3]/3 + (x^3*Sin[x^3])/3

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Mathics [A]
time = 2.38, size = 16, normalized size = 0.80 \begin {gather*} \frac {x^3 \text {Sin}\left [x^3\right ]}{3}+\frac {\text {Cos}\left [x^3\right ]}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^5*Cos[x^3],x]')

[Out]

x ^ 3 Sin[x ^ 3] / 3 + Cos[x ^ 3] / 3

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Maple [A]
time = 0.04, size = 17, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {\cos \left (x^{3}\right )}{3}+\frac {x^{3} \sin \left (x^{3}\right )}{3}\)	\(17\)
default	\(\frac {\cos \left (x^{3}\right )}{3}+\frac {x^{3} \sin \left (x^{3}\right )}{3}\)	\(17\)
risch	\(\frac {\cos \left (x^{3}\right )}{3}+\frac {x^{3} \sin \left (x^{3}\right )}{3}\)	\(17\)
norman	\(\frac {\frac {2 x^{3} \tan \left (\frac {x^{3}}{2}\right )}{3}+\frac {2}{3}}{1+\tan ^{2}\left (\frac {x^{3}}{2}\right )}\)	\(27\)
meijerg	\(\frac {2 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (x^{3}\right )}{2 \sqrt {\pi }}+\frac {x^{3} \sin \left (x^{3}\right )}{2 \sqrt {\pi }}\right )}{3}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cos(x^3),x,method=_RETURNVERBOSE)

[Out]

1/3*cos(x^3)+1/3*x^3*sin(x^3)

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} \sin \left (x^{3}\right ) + \frac {1}{3} \, \cos \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cos(x^3),x, algorithm="maxima")

[Out]

1/3*x^3*sin(x^3) + 1/3*cos(x^3)

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Fricas [A]
time = 0.36, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} \sin \left (x^{3}\right ) + \frac {1}{3} \, \cos \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cos(x^3),x, algorithm="fricas")

[Out]

1/3*x^3*sin(x^3) + 1/3*cos(x^3)

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Sympy [A]
time = 0.32, size = 15, normalized size = 0.75 \begin {gather*} \frac {x^{3} \sin {\left (x^{3} \right )}}{3} + \frac {\cos {\left (x^{3} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*cos(x**3),x)

[Out]

x**3*sin(x**3)/3 + cos(x**3)/3

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Giac [A]
time = 0.00, size = 16, normalized size = 0.80 \begin {gather*} \frac {\cos \left (x^{3}\right )+x^{3} \sin \left (x^{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cos(x^3),x)

[Out]

1/3*x^3*sin(x^3) + 1/3*cos(x^3)

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Mupad [B]
time = 0.19, size = 16, normalized size = 0.80 \begin {gather*} \frac {\cos \left (x^3\right )}{3}+\frac {x^3\,\sin \left (x^3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cos(x^3),x)

[Out]

cos(x^3)/3 + (x^3*sin(x^3))/3

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