3.764 \(\int x (\cos ^3(x^2)-\sin ^3(x^2)) \, dx\)

Optimal. Leaf size=37 \[ -\frac{1}{6} \sin ^3\left (x^2\right )+\frac{\sin \left (x^2\right )}{2}-\frac{1}{6} \cos ^3\left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]

[Out]

Cos[x^2]/2 - Cos[x^2]^3/6 + Sin[x^2]/2 - Sin[x^2]^3/6

________________________________________________________________________________________

Rubi [A]  time = 0.0335579, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {14, 3380, 2633, 3379} \[ -\frac{1}{6} \sin ^3\left (x^2\right )+\frac{\sin \left (x^2\right )}{2}-\frac{1}{6} \cos ^3\left (x^2\right )+\frac{\cos \left (x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]

Int[x*(Cos[x^2]^3 - Sin[x^2]^3),x]

[Out]

Cos[x^2]/2 - Cos[x^2]^3/6 + Sin[x^2]/2 - Sin[x^2]^3/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[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 \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx &=\int \left (x \cos ^3\left (x^2\right )-x \sin ^3\left (x^2\right )\right ) \, dx\\ &=\int x \cos ^3\left (x^2\right ) \, dx-\int x \sin ^3\left (x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \cos ^3(x) \, dx,x,x^2\right )-\frac{1}{2} \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (x^2\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (x^2\right )\right )\\ &=\frac{\cos \left (x^2\right )}{2}-\frac{1}{6} \cos ^3\left (x^2\right )+\frac{\sin \left (x^2\right )}{2}-\frac{1}{6} \sin ^3\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0230532, size = 37, normalized size = 1. \[ -\frac{1}{6} \sin ^3\left (x^2\right )+\frac{\sin \left (x^2\right )}{2}+\frac{3 \cos \left (x^2\right )}{8}-\frac{1}{24} \cos \left (3 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(Cos[x^2]^3 - Sin[x^2]^3),x]

[Out]

(3*Cos[x^2])/8 - Cos[3*x^2]/24 + Sin[x^2]/2 - Sin[x^2]^3/6

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 30, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ({x}^{2} \right ) \right ) ^{2} \right ) \sin \left ({x}^{2} \right ) }{6}}+{\frac{ \left ( 2+ \left ( \sin \left ({x}^{2} \right ) \right ) ^{2} \right ) \cos \left ({x}^{2} \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(cos(x^2)^3-sin(x^2)^3),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.968764, size = 39, normalized size = 1.05 \begin{align*} -\frac{1}{24} \, \cos \left (3 \, x^{2}\right ) + \frac{3}{8} \, \cos \left (x^{2}\right ) + \frac{1}{24} \, \sin \left (3 \, x^{2}\right ) + \frac{3}{8} \, \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(cos(x^2)^3-sin(x^2)^3),x, algorithm="maxima")

[Out]

-1/24*cos(3*x^2) + 3/8*cos(x^2) + 1/24*sin(3*x^2) + 3/8*sin(x^2)

________________________________________________________________________________________

Fricas [A]  time = 2.08791, size = 86, normalized size = 2.32 \begin{align*} -\frac{1}{6} \, \cos \left (x^{2}\right )^{3} + \frac{1}{6} \,{\left (\cos \left (x^{2}\right )^{2} + 2\right )} \sin \left (x^{2}\right ) + \frac{1}{2} \, \cos \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(cos(x^2)^3-sin(x^2)^3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.609055, size = 42, normalized size = 1.14 \begin{align*} \frac{\sin ^{3}{\left (x^{2} \right )}}{3} + \frac{\sin ^{2}{\left (x^{2} \right )} \cos{\left (x^{2} \right )}}{2} + \frac{\sin{\left (x^{2} \right )} \cos ^{2}{\left (x^{2} \right )}}{2} + \frac{\cos ^{3}{\left (x^{2} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(cos(x**2)**3-sin(x**2)**3),x)

[Out]

sin(x**2)**3/3 + sin(x**2)**2*cos(x**2)/2 + sin(x**2)*cos(x**2)**2/2 + cos(x**2)**3/3

________________________________________________________________________________________

Giac [A]  time = 1.07506, size = 39, normalized size = 1.05 \begin{align*} -\frac{1}{6} \, \cos \left (x^{2}\right )^{3} - \frac{1}{6} \, \sin \left (x^{2}\right )^{3} + \frac{1}{2} \, \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(cos(x^2)^3-sin(x^2)^3),x, algorithm="giac")

[Out]

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