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]

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

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Rubi [A]  time = 0.04, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {1}{6} \tan ^3\left (x^2\right ) \]

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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fricas [B]  time = 0.43, size = 20, normalized size = 2.00 \[ -\frac {{\left (\cos \left (x^{2}\right )^{2} - 1\right )} \sin \left (x^{2}\right )}{6 \, \cos \left (x^{2}\right )^{3}} \]

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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giac [A]  time = 0.14, size = 8, normalized size = 0.80 \[ \frac {1}{6} \, \tan \left (x^{2}\right )^{3} \]

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

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maple [A]  time = 0.10, size = 15, normalized size = 1.50 \[ \frac {\sin ^{3}\left (x^{2}\right )}{6 \cos \left (x^{2}\right )^{3}} \]

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 = 0.32, size = 8, normalized size = 0.80 \[ \frac {1}{6} \, \tan \left (x^{2}\right )^{3} \]

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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mupad [B]  time = 3.09, size = 19, normalized size = 1.90 \[ \frac {\mathrm {tan}\left (x^2\right )}{6\,{\cos \left (x^2\right )}^2}-\frac {\mathrm {tan}\left (x^2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*tan(x^2)^2)/cos(x^2)^2,x)

[Out]

tan(x^2)/(6*cos(x^2)^2) - tan(x^2)/6

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sympy [A]  time = 1.04, size = 7, normalized size = 0.70 \[ \frac {\tan ^{3}{\left (x^{2} \right )}}{6} \]

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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