3.824 \(\int 4 x \tan (x^2) \, dx\)

Optimal. Leaf size=7 \[ -2 \log \left (\cos \left (x^2\right )\right ) \]

[Out]

-2*Log[Cos[x^2]]

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Rubi [A]  time = 0.0072001, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 3747, 3475} \[ -2 \log \left (\cos \left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Log[Cos[x^2]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3747

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Tan[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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int 4 x \tan \left (x^2\right ) \, dx &=4 \int x \tan \left (x^2\right ) \, dx\\ &=2 \operatorname{Subst}\left (\int \tan (x) \, dx,x,x^2\right )\\ &=-2 \log \left (\cos \left (x^2\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0058107, size = 7, normalized size = 1. \[ -2 \log \left (\cos \left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Log[Cos[x^2]]

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Maple [A]  time = 0.003, size = 8, normalized size = 1.1 \begin{align*} -2\,\ln \left ( \cos \left ({x}^{2} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(cos(x^2))

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Maxima [A]  time = 0.961662, size = 9, normalized size = 1.29 \begin{align*} 2 \, \log \left (\sec \left (x^{2}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sec(x^2))

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Fricas [A]  time = 2.27923, size = 35, normalized size = 5. \begin{align*} -\log \left (\frac{1}{\tan \left (x^{2}\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(1/(tan(x^2)^2 + 1))

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Sympy [A]  time = 0.127445, size = 8, normalized size = 1.14 \begin{align*} \log{\left (\tan ^{2}{\left (x^{2} \right )} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x**2)**2 + 1)

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Giac [A]  time = 1.06695, size = 12, normalized size = 1.71 \begin{align*} \log \left (\tan \left (x^{2}\right )^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x^2)^2 + 1)