3.768 \(\int x^3 \cos (x^4) \, dx\)

Optimal. Leaf size=8 \[ \frac {\sin \left (x^4\right )}{4} \]

[Out]

1/4*sin(x^4)

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3380, 2637} \[ \frac {\sin \left (x^4\right )}{4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Cos[x^4],x]

[Out]

Sin[x^4]/4

Rule 2637

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

Rule 3380

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^3 \cos \left (x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \cos (x) \, dx,x,x^4\right )\\ &=\frac {\sin \left (x^4\right )}{4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\sin \left (x^4\right )}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cos[x^4],x]

[Out]

Sin[x^4]/4

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fricas [A]  time = 0.67, size = 6, normalized size = 0.75 \[ \frac {1}{4} \, \sin \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cos(x^4),x, algorithm="fricas")

[Out]

1/4*sin(x^4)

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giac [A]  time = 0.14, size = 6, normalized size = 0.75 \[ \frac {1}{4} \, \sin \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cos(x^4),x, algorithm="giac")

[Out]

1/4*sin(x^4)

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maple [A]  time = 0.04, size = 7, normalized size = 0.88 \[ \frac {\sin \left (x^{4}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cos(x^4),x)

[Out]

1/4*sin(x^4)

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maxima [A]  time = 0.31, size = 6, normalized size = 0.75 \[ \frac {1}{4} \, \sin \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cos(x^4),x, algorithm="maxima")

[Out]

1/4*sin(x^4)

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mupad [B]  time = 0.07, size = 6, normalized size = 0.75 \[ \frac {\sin \left (x^4\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cos(x^4),x)

[Out]

sin(x^4)/4

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sympy [A]  time = 0.48, size = 5, normalized size = 0.62 \[ \frac {\sin {\left (x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cos(x**4),x)

[Out]

sin(x**4)/4

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