3.753 \(\int x \cos ^4(x^2) \, dx\)

Optimal. Leaf size=34 \[ \frac{3 x^2}{16}+\frac{1}{8} \sin \left (x^2\right ) \cos ^3\left (x^2\right )+\frac{3}{16} \sin \left (x^2\right ) \cos \left (x^2\right ) \]

[Out]

(3*x^2)/16 + (3*Cos[x^2]*Sin[x^2])/16 + (Cos[x^2]^3*Sin[x^2])/8

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Rubi [A]  time = 0.0224351, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3380, 2635, 8} \[ \frac{3 x^2}{16}+\frac{1}{8} \sin \left (x^2\right ) \cos ^3\left (x^2\right )+\frac{3}{16} \sin \left (x^2\right ) \cos \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[x^2]^4,x]

[Out]

(3*x^2)/16 + (3*Cos[x^2]*Sin[x^2])/16 + (Cos[x^2]^3*Sin[x^2])/8

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x \cos ^4\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cos ^4(x) \, dx,x,x^2\right )\\ &=\frac{1}{8} \cos ^3\left (x^2\right ) \sin \left (x^2\right )+\frac{3}{8} \operatorname{Subst}\left (\int \cos ^2(x) \, dx,x,x^2\right )\\ &=\frac{3}{16} \cos \left (x^2\right ) \sin \left (x^2\right )+\frac{1}{8} \cos ^3\left (x^2\right ) \sin \left (x^2\right )+\frac{3}{16} \operatorname{Subst}\left (\int 1 \, dx,x,x^2\right )\\ &=\frac{3 x^2}{16}+\frac{3}{16} \cos \left (x^2\right ) \sin \left (x^2\right )+\frac{1}{8} \cos ^3\left (x^2\right ) \sin \left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0167307, size = 28, normalized size = 0.82 \[ \frac{3 x^2}{16}+\frac{1}{8} \sin \left (2 x^2\right )+\frac{1}{64} \sin \left (4 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[x^2]^4,x]

[Out]

(3*x^2)/16 + Sin[2*x^2]/8 + Sin[4*x^2]/64

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Maple [A]  time = 0.01, size = 26, normalized size = 0.8 \begin{align*}{\frac{\sin \left ({x}^{2} \right ) }{8} \left ( \left ( \cos \left ({x}^{2} \right ) \right ) ^{3}+{\frac{3\,\cos \left ({x}^{2} \right ) }{2}} \right ) }+{\frac{3\,{x}^{2}}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x^2)^4,x)

[Out]

1/8*(cos(x^2)^3+3/2*cos(x^2))*sin(x^2)+3/16*x^2

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Maxima [A]  time = 0.957417, size = 30, normalized size = 0.88 \begin{align*} \frac{3}{16} \, x^{2} + \frac{1}{64} \, \sin \left (4 \, x^{2}\right ) + \frac{1}{8} \, \sin \left (2 \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x^2)^4,x, algorithm="maxima")

[Out]

3/16*x^2 + 1/64*sin(4*x^2) + 1/8*sin(2*x^2)

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Fricas [A]  time = 2.02743, size = 73, normalized size = 2.15 \begin{align*} \frac{3}{16} \, x^{2} + \frac{1}{16} \,{\left (2 \, \cos \left (x^{2}\right )^{3} + 3 \, \cos \left (x^{2}\right )\right )} \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x^2)^4,x, algorithm="fricas")

[Out]

3/16*x^2 + 1/16*(2*cos(x^2)^3 + 3*cos(x^2))*sin(x^2)

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Sympy [B]  time = 1.20831, size = 76, normalized size = 2.24 \begin{align*} \frac{3 x^{2} \sin ^{4}{\left (x^{2} \right )}}{16} + \frac{3 x^{2} \sin ^{2}{\left (x^{2} \right )} \cos ^{2}{\left (x^{2} \right )}}{8} + \frac{3 x^{2} \cos ^{4}{\left (x^{2} \right )}}{16} + \frac{3 \sin ^{3}{\left (x^{2} \right )} \cos{\left (x^{2} \right )}}{16} + \frac{5 \sin{\left (x^{2} \right )} \cos ^{3}{\left (x^{2} \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x**2)**4,x)

[Out]

3*x**2*sin(x**2)**4/16 + 3*x**2*sin(x**2)**2*cos(x**2)**2/8 + 3*x**2*cos(x**2)**4/16 + 3*sin(x**2)**3*cos(x**2
)/16 + 5*sin(x**2)*cos(x**2)**3/16

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Giac [A]  time = 1.08616, size = 30, normalized size = 0.88 \begin{align*} \frac{3}{16} \, x^{2} + \frac{1}{64} \, \sin \left (4 \, x^{2}\right ) + \frac{1}{8} \, \sin \left (2 \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x^2)^4,x, algorithm="giac")

[Out]

3/16*x^2 + 1/64*sin(4*x^2) + 1/8*sin(2*x^2)