3.924 \(\int x \sec ^2(x^2) \tan ^2(x^2) \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{6} \tan ^3\left (x^2\right ) \]

[Out]

Tan[x^2]^3/6

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Rubi [A]  time = 0.0369431, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {6686} \[ \frac{1}{6} \tan ^3\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x^2]^3/6

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x \sec ^2\left (x^2\right ) \tan ^2\left (x^2\right ) \, dx &=\frac{1}{6} \tan ^3\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.00331, size = 10, normalized size = 1. \[ \frac{1}{6} \tan ^3\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x^2]^3/6

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Maple [A]  time = 0.022, size = 15, normalized size = 1.5 \begin{align*}{\frac{ \left ( \sin \left ({x}^{2} \right ) \right ) ^{3}}{6\, \left ( \cos \left ({x}^{2} \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.935444, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{6} \, \tan \left (x^{2}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*tan(x^2)^3

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Fricas [B]  time = 2.05899, size = 58, normalized size = 5.8 \begin{align*} -\frac{{\left (\cos \left (x^{2}\right )^{2} - 1\right )} \sin \left (x^{2}\right )}{6 \, \cos \left (x^{2}\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.73092, size = 7, normalized size = 0.7 \begin{align*} \frac{\tan ^{3}{\left (x^{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x**2)**3/6

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Giac [A]  time = 1.06285, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{6} \, \tan \left (x^{2}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*tan(x^2)^3