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

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

[Out]

-ln(ln(cos(x)))

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Rubi [A]  time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, 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 29

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

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 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])

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*}

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Mathematica [A]  time = 0.01, size = 6, normalized size = 1.00 \[ -\log (\log (\cos (x))) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[Log[Cos[x]]]

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fricas [A]  time = 0.43, size = 6, normalized size = 1.00 \[ -\log \left (\log \left (\cos \relax (x)\right )\right ) \]

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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giac [A]  time = 0.16, size = 7, normalized size = 1.17 \[ -\log \left ({\left | \log \left (\cos \relax (x)\right ) \right |}\right ) \]

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))))

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maple [A]  time = 0.30, size = 7, normalized size = 1.17 \[ -\ln \left (\ln \left (\cos \relax (x )\right )\right ) \]

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 = 0.44, size = 6, normalized size = 1.00 \[ -\log \left (\log \left (\cos \relax (x)\right )\right ) \]

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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mupad [B]  time = 0.40, size = 6, normalized size = 1.00 \[ -\ln \left (\ln \left (\cos \relax (x)\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(cos(x)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\relax (x )}}{\log {\left (\cos {\relax (x )} \right )}}\, dx \]

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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