3.186 $$\int \frac{\tan (x)}{\log (\cos (x))} \, dx$$

Optimal. Leaf size=6 $-\log (\log (\cos (x)))$

[Out]

-Log[Log[Cos[x]]]

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Rubi [A]  time = 0.0215558, antiderivative size = 6, 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 = {4339, 2302, 29} $-\log (\log (\cos (x)))$

Antiderivative was successfully veriﬁed.

[In]

Int[Tan[x]/Log[Cos[x]],x]

[Out]

-Log[Log[Cos[x]]]

Rule 4339

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*
c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0083844, size = 6, normalized size = 1. $-\log (\log (\cos (x)))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tan[x]/Log[Cos[x]],x]

[Out]

-Log[Log[Cos[x]]]

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Maple [A]  time = 0.007, size = 7, normalized size = 1.2 \begin{align*} -\ln \left ( \ln \left ( \cos \left ( x \right ) \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/ln(cos(x)),x)

[Out]

-ln(ln(cos(x)))

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Maxima [A]  time = 0.998179, size = 8, normalized size = 1.33 \begin{align*} -\log \left (\log \left (\cos \left (x\right )\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/log(cos(x)),x, algorithm="maxima")

[Out]

-log(log(cos(x)))

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Fricas [A]  time = 2.1658, size = 26, normalized size = 4.33 \begin{align*} -\log \left (\log \left (\cos \left (x\right )\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/log(cos(x)),x, algorithm="fricas")

[Out]

-log(log(cos(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (x \right )}}{\log{\left (\cos{\left (x \right )} \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/ln(cos(x)),x)

[Out]

Integral(tan(x)/log(cos(x)), x)

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Giac [A]  time = 1.3045, size = 9, normalized size = 1.5 \begin{align*} -\log \left ({\left | \log \left (\cos \left (x\right )\right ) \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/log(cos(x)),x, algorithm="giac")

[Out]

-log(abs(log(cos(x))))