3.301 \(\int \csc (x) \log (\tan (x)) \sec (x) \, dx\)

Optimal. Leaf size=9 \[ \frac {1}{2} \log ^2(\tan (x)) \]

[Out]

1/2*ln(tan(x))^2

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2620, 29, 6686} \[ \frac {1}{2} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]*Log[Tan[x]]*Sec[x],x]

[Out]

Log[Tan[x]]^2/2

Rule 29

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

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{2} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]*Log[Tan[x]]*Sec[x],x]

[Out]

Log[Tan[x]]^2/2

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fricas [A]  time = 0.43, size = 12, normalized size = 1.33 \[ \frac {1}{2} \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(sin(x)/cos(x))^2

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giac [A]  time = 1.08, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\tan \relax (x)\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(tan(x))^2

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maple [A]  time = 0.04, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\tan \relax (x )\right )^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(tan(x))^2

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maxima [A]  time = 0.50, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\tan \relax (x)\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(tan(x))^2

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mupad [B]  time = 2.59, size = 27, normalized size = 3.00 \[ \frac {{\ln \left (-\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}+1}\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1))^2/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(tan(x))/(sin(x)*cos(x)), x)

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