\(\int \frac {\cos (\sqrt [6]{x})}{x^{5/6}} \, dx\) [62]

Optimal result

Integrand size = 12, antiderivative size = 8 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \sin \left (\sqrt [6]{x}\right ) \]

[Out]

6*sin(x^(1/6))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3461, 2717} \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \sin \left (\sqrt [6]{x}\right ) \]

[In]

Int[Cos[x^(1/6)]/x^(5/6),x]

[Out]

6*Sin[x^(1/6)]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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*} \text {integral}& = 6 \text {Subst}\left (\int \cos (x) \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sin \left (\sqrt [6]{x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \sin \left (\sqrt [6]{x}\right ) \]

[In]

Integrate[Cos[x^(1/6)]/x^(5/6),x]

[Out]

6*Sin[x^(1/6)]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(6 \sin \left (x^{\frac {1}{6}}\right )\)	\(7\)
default	\(6 \sin \left (x^{\frac {1}{6}}\right )\)	\(7\)
meijerg	\(6 \sin \left (x^{\frac {1}{6}}\right )\)	\(7\)

[In]

int(cos(x^(1/6))/x^(5/6),x,method=_RETURNVERBOSE)

[Out]

6*sin(x^(1/6))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \, \sin \left (x^{\frac {1}{6}}\right ) \]

[In]

integrate(cos(x^(1/6))/x^(5/6),x, algorithm="fricas")

[Out]

6*sin(x^(1/6))

Sympy [A] (verification not implemented)

Time = 6.71 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \sin {\left (\sqrt [6]{x} \right )} \]

[In]

integrate(cos(x**(1/6))/x**(5/6),x)

[Out]

6*sin(x**(1/6))

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \, \sin \left (x^{\frac {1}{6}}\right ) \]

[In]

integrate(cos(x^(1/6))/x^(5/6),x, algorithm="maxima")

[Out]

6*sin(x^(1/6))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6 \, \sin \left (x^{\frac {1}{6}}\right ) \]

[In]

integrate(cos(x^(1/6))/x^(5/6),x, algorithm="giac")

[Out]

6*sin(x^(1/6))

Mupad [B] (verification not implemented)

Time = 13.59 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cos \left (\sqrt [6]{x}\right )}{x^{5/6}} \, dx=6\,\sin \left (x^{1/6}\right ) \]

[In]

int(cos(x^(1/6))/x^(5/6),x)

[Out]

6*sin(x^(1/6))