\(\int \frac {\tan (x)}{\log (\cos (x))} \, dx\) [186]

Optimal result

Integrand size = 8, antiderivative size = 6 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log (\log (\cos (x))) \]

[Out]

-ln(ln(cos(x)))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4424, 2339, 29} \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log (\log (\cos (x))) \]

[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 2339

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 4424

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log (\log (\cos (x))) \]

[In]

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

[Out]

-Log[Log[Cos[x]]]

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(-\ln \left (\ln \left (\cos \left (x \right )\right )\right )\)	\(7\)
default	\(-\ln \left (\ln \left (\cos \left (x \right )\right )\right )\)	\(7\)
risch	\(-\ln \left (\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i \cos \left (x \right )\right )^{3}}{2}+\ln \left (2\right )-\ln \left (1+{\mathrm e}^{2 i x}\right )+\ln \left ({\mathrm e}^{i x}\right )\right )\)	\(109\)

[In]

int(tan(x)/ln(cos(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(cos(x)))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log \left (\log \left (\cos \left (x\right )\right )\right ) \]

[In]

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

[Out]

-log(log(cos(x)))

Sympy [F]

\[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=\int \frac {\tan {\left (x \right )}}{\log {\left (\cos {\left (x \right )} \right )}}\, dx \]

[In]

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

[Out]

Integral(tan(x)/log(cos(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log \left (\log \left (\cos \left (x\right )\right )\right ) \]

[In]

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

[Out]

-log(log(cos(x)))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log \left ({\left | \log \left (\cos \left (x\right )\right ) \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.59 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\ln \left (\ln \left (\cos \left (x\right )\right )\right ) \]

[In]

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

[Out]

-log(log(cos(x)))