3.761 \(\int x \sec ^2(x^2) \, dx\)

Optimal. Leaf size=8 \[ \frac{\tan \left (x^2\right )}{2} \]

[Out]

Tan[x^2]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0130906, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4204, 3767, 8} \[ \frac{\tan \left (x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sec[x^2]^2,x]

[Out]

Tan[x^2]/2

Rule 4204

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 3767

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

Rule 8

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

Rubi steps

\begin{align*} \int x \sec ^2\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sec ^2(x) \, dx,x,x^2\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int 1 \, dx,x,-\tan \left (x^2\right )\right )\right )\\ &=\frac{\tan \left (x^2\right )}{2}\\ \end{align*}

Mathematica [A]  time = 0.0163709, size = 8, normalized size = 1. \[ \frac{\tan \left (x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sec[x^2]^2,x]

[Out]

Tan[x^2]/2

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 7, normalized size = 0.9 \begin{align*}{\frac{\tan \left ({x}^{2} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sec(x^2)^2,x)

[Out]

1/2*tan(x^2)

________________________________________________________________________________________

Maxima [B]  time = 0.964929, size = 47, normalized size = 5.88 \begin{align*} \frac{\sin \left (2 \, x^{2}\right )}{\cos \left (2 \, x^{2}\right )^{2} + \sin \left (2 \, x^{2}\right )^{2} + 2 \, \cos \left (2 \, x^{2}\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.97484, size = 31, normalized size = 3.88 \begin{align*} \frac{\sin \left (x^{2}\right )}{2 \, \cos \left (x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sec ^{2}{\left (x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sec(x**2)**2,x)

[Out]

Integral(x*sec(x**2)**2, x)

________________________________________________________________________________________

Giac [A]  time = 1.07706, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \tan \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*tan(x^2)