3.1.100 \(\int \sin (\sqrt [3]{x}) \, dx\) [100]

Optimal. Leaf size=35 \[ 6 \cos \left (\sqrt [3]{x}\right )-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right ) \]

[Out]

6*cos(x^(1/3))-3*x^(2/3)*cos(x^(1/3))+6*x^(1/3)*sin(x^(1/3))

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3442, 3377, 2718} \begin {gather*} -3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )+6 \cos \left (\sqrt [3]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x^(1/3)],x]

[Out]

6*Cos[x^(1/3)] - 3*x^(2/3)*Cos[x^(1/3)] + 6*x^(1/3)*Sin[x^(1/3)]

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 3442

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=3 \text {Subst}\left (\int x^2 \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \text {Subst}\left (\int x \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-6 \text {Subst}\left (\int \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=6 \cos \left (\sqrt [3]{x}\right )-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.83 \begin {gather*} -3 \left (-2+x^{2/3}\right ) \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x^(1/3)],x]

[Out]

-3*(-2 + x^(2/3))*Cos[x^(1/3)] + 6*x^(1/3)*Sin[x^(1/3)]

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Maple [A]
time = 0.02, size = 26, normalized size = 0.74

method	result	size
derivativedivides	\(6 \cos \left (x^{\frac {1}{3}}\right )-3 x^{\frac {2}{3}} \cos \left (x^{\frac {1}{3}}\right )+6 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )\)	\(26\)
default	\(6 \cos \left (x^{\frac {1}{3}}\right )-3 x^{\frac {2}{3}} \cos \left (x^{\frac {1}{3}}\right )+6 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )\)	\(26\)
meijerg	\(12 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{\frac {2}{3}}}{2}+1\right ) \cos \left (x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}+\frac {x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}\right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x^(1/3)),x,method=_RETURNVERBOSE)

[Out]

6*cos(x^(1/3))-3*x^(2/3)*cos(x^(1/3))+6*x^(1/3)*sin(x^(1/3))

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Maxima [A]
time = 0.37, size = 21, normalized size = 0.60 \begin {gather*} -3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3)),x, algorithm="maxima")

[Out]

-3*(x^(2/3) - 2)*cos(x^(1/3)) + 6*x^(1/3)*sin(x^(1/3))

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Fricas [A]
time = 0.59, size = 21, normalized size = 0.60 \begin {gather*} -3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3)),x, algorithm="fricas")

[Out]

-3*(x^(2/3) - 2)*cos(x^(1/3)) + 6*x^(1/3)*sin(x^(1/3))

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Sympy [A]
time = 0.17, size = 34, normalized size = 0.97 \begin {gather*} - 3 x^{\frac {2}{3}} \cos {\left (\sqrt [3]{x} \right )} + 6 \sqrt [3]{x} \sin {\left (\sqrt [3]{x} \right )} + 6 \cos {\left (\sqrt [3]{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x**(1/3)),x)

[Out]

-3*x**(2/3)*cos(x**(1/3)) + 6*x**(1/3)*sin(x**(1/3)) + 6*cos(x**(1/3))

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Giac [A]
time = 0.43, size = 21, normalized size = 0.60 \begin {gather*} -3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/3)),x, algorithm="giac")

[Out]

-3*(x^(2/3) - 2)*cos(x^(1/3)) + 6*x^(1/3)*sin(x^(1/3))

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Mupad [B]
time = 0.21, size = 21, normalized size = 0.60 \begin {gather*} 6\,x^{1/3}\,\sin \left (x^{1/3}\right )-3\,\cos \left (x^{1/3}\right )\,\left (x^{2/3}-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x^(1/3)),x)

[Out]

6*x^(1/3)*sin(x^(1/3)) - 3*cos(x^(1/3))*(x^(2/3) - 2)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Antiderivative was successfully verified.

[In]

int(sin(x^(1/3)),x)

[Out]

not solved

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