3.1.81 \(\int \log (\cos (x)) \sec ^2(x) \, dx\) [81]

Optimal. Leaf size=12 \[ -x+\tan (x)+\log (\cos (x)) \tan (x) \]

[Out]

-x+tan(x)+ln(cos(x))*tan(x)

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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3852, 8, 2634, 3554} \begin {gather*} -x+\tan (x)+\tan (x) \log (\cos (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[Cos[x]]*Sec[x]^2,x]

[Out]

-x + Tan[x] + Log[Cos[x]]*Tan[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \log (\cos (x)) \sec ^2(x) \, dx &=\log (\cos (x)) \tan (x)+\int \tan ^2(x) \, dx\\ &=\tan (x)+\log (\cos (x)) \tan (x)-\int 1 \, dx\\ &=-x+\tan (x)+\log (\cos (x)) \tan (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -x+\tan (x)+\log (\cos (x)) \tan (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[Cos[x]]*Sec[x]^2,x]

[Out]

-x + Tan[x] + Log[Cos[x]]*Tan[x]

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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 67, normalized size = 5.58

method	result	size
norman	\(\frac {x -x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) \ln \left (\frac {1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\right )-2 \tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}\)	\(57\)
default	\(-4 i \left (\frac {\frac {{\mathrm e}^{2 i x} \ln \left (\left ({\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )}{2}-\frac {1}{2}}{{\mathrm e}^{2 i x}+1}-\frac {\ln \left ({\mathrm e}^{2 i x}+1\right )}{4}+\frac {\ln \left (2\right )}{2 \,{\mathrm e}^{2 i x}+2}\right )\)	\(67\)
risch	\(-\frac {2 i \ln \left ({\mathrm e}^{i x}\right )}{{\mathrm e}^{2 i x}+1}+\frac {\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}+1\right )\right ) \mathrm {csgn}\left (i \cos \left (x \right )\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (i \cos \left (x \right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}+1\right )\right ) \mathrm {csgn}\left (i \cos \left (x \right )\right )^{2}+\pi \mathrm {csgn}\left (i \cos \left (x \right )\right )^{3}-i \ln \left ({\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{2 i x}-2 x \,{\mathrm e}^{2 i x}-2 i \ln \left (2\right )+i \ln \left ({\mathrm e}^{2 i x}+1\right )+2 i-2 x}{{\mathrm e}^{2 i x}+1}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(cos(x))*sec(x)^2,x,method=_RETURNVERBOSE)

[Out]

-4*I*((1/2*exp(2*I*x)*ln((exp(I*x)^2+1)/exp(I*x))-1/2)/(exp(2*I*x)+1)-1/4*ln(exp(2*I*x)+1)+1/2*ln(2)/(exp(I*x)
^2+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (12) = 24\).
time = 1.13, size = 94, normalized size = 7.83 \begin {gather*} -\frac {2 \, \log \left (-\frac {\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}\right ) \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}} - \frac {2 \, \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}} - 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cos(x))*sec(x)^2,x, algorithm="maxima")

[Out]

-2*log(-(sin(x)^2/(cos(x) + 1)^2 - 1)/(sin(x)^2/(cos(x) + 1)^2 + 1))*sin(x)/((sin(x)^2/(cos(x) + 1)^2 - 1)*(co
s(x) + 1)) - 2*sin(x)/((sin(x)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)) - 2*arctan(sin(x)/(cos(x) + 1))

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Fricas [A]
time = 0.63, size = 22, normalized size = 1.83 \begin {gather*} -\frac {x \cos \left (x\right ) - \log \left (\cos \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cos(x))*sec(x)^2,x, algorithm="fricas")

[Out]

-(x*cos(x) - log(cos(x))*sin(x) - sin(x))/cos(x)

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Sympy [A]
time = 20.40, size = 15, normalized size = 1.25 \begin {gather*} - x + \log {\left (\cos {\left (x \right )} \right )} \tan {\left (x \right )} + \frac {\sin {\left (x \right )}}{\cos {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(cos(x))*sec(x)**2,x)

[Out]

-x + log(cos(x))*tan(x) + sin(x)/cos(x)

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Giac [A]
time = 0.65, size = 12, normalized size = 1.00 \begin {gather*} \log \left (\cos \left (x\right )\right ) \tan \left (x\right ) - x + \tan \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(cos(x))*sec(x)^2,x, algorithm="giac")

[Out]

log(cos(x))*tan(x) - x + tan(x)

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Mupad [B]
time = 0.43, size = 35, normalized size = 2.92 \begin {gather*} \mathrm {tan}\left (x\right )-2\,x+\ln \left (\cos \left (x\right )\right )\,\mathrm {tan}\left (x\right )+\ln \left (\cos \left (x\right )\right )\,1{}\mathrm {i}-\ln \left (\cos \left (2\,x\right )+1+\sin \left (2\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(cos(x))/cos(x)^2,x)

[Out]

log(cos(x))*1i - 2*x - log(cos(2*x) + sin(2*x)*1i + 1)*1i + tan(x) + log(cos(x))*tan(x)

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