3.783 \(\int (1+x^2 \cos (x^3)) \, dx\)

Optimal. Leaf size=10 \[ \frac{\sin \left (x^3\right )}{3}+x \]

[Out]

x + Sin[x^3]/3

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Rubi [A]  time = 0.0098915, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3380, 2637} \[ \frac{\sin \left (x^3\right )}{3}+x \]

Antiderivative was successfully verified.

[In]

Int[1 + x^2*Cos[x^3],x]

[Out]

x + Sin[x^3]/3

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]))

Rule 2637

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

Rubi steps

\begin{align*} \int \left (1+x^2 \cos \left (x^3\right )\right ) \, dx &=x+\int x^2 \cos \left (x^3\right ) \, dx\\ &=x+\frac{1}{3} \operatorname{Subst}\left (\int \cos (x) \, dx,x,x^3\right )\\ &=x+\frac{\sin \left (x^3\right )}{3}\\ \end{align*}

Mathematica [A]  time = 0.0027596, size = 10, normalized size = 1. \[ \frac{\sin \left (x^3\right )}{3}+x \]

Antiderivative was successfully verified.

[In]

Integrate[1 + x^2*Cos[x^3],x]

[Out]

x + Sin[x^3]/3

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Maple [A]  time = 0.003, size = 9, normalized size = 0.9 \begin{align*} x+{\frac{\sin \left ({x}^{3} \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1+x^2*cos(x^3),x)

[Out]

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

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Maxima [A]  time = 0.949289, size = 11, normalized size = 1.1 \begin{align*} x + \frac{1}{3} \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+x^2*cos(x^3),x, algorithm="maxima")

[Out]

x + 1/3*sin(x^3)

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Fricas [A]  time = 2.05855, size = 24, normalized size = 2.4 \begin{align*} x + \frac{1}{3} \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+x^2*cos(x^3),x, algorithm="fricas")

[Out]

x + 1/3*sin(x^3)

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Sympy [A]  time = 0.297757, size = 7, normalized size = 0.7 \begin{align*} x + \frac{\sin{\left (x^{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+x**2*cos(x**3),x)

[Out]

x + sin(x**3)/3

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Giac [A]  time = 1.0812, size = 11, normalized size = 1.1 \begin{align*} x + \frac{1}{3} \, \sin \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+x^2*cos(x^3),x, algorithm="giac")

[Out]

x + 1/3*sin(x^3)