3.750 \(\int \csc (2 x) \log (\tan (x)) \, dx\)

Optimal. Leaf size=9 \[ \frac {1}{4} \log ^2(\tan (x)) \]

[Out]

1/4*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 = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3770, 6686} \[ \frac {1}{4} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[2*x]*Log[Tan[x]],x]

[Out]

Log[Tan[x]]^2/4

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 (2 x) \log (\tan (x)) \, dx &=\frac {1}{4} \log ^2(\tan (x))\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{4} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*x]*Log[Tan[x]],x]

[Out]

Log[Tan[x]]^2/4

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fricas [A]  time = 0.51, size = 7, normalized size = 0.78 \[ \frac {1}{4} \, \log \left (\tan \relax (x)\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*log(tan(x)),x, algorithm="fricas")

[Out]

1/4*log(tan(x))^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (2 \, x\right ) \log \left (\tan \relax (x)\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*log(tan(x)),x, algorithm="giac")

[Out]

integrate(csc(2*x)*log(tan(x)), x)

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maple [A]  time = 0.06, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\tan \relax (x )\right )^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2*x)*ln(tan(x)),x)

[Out]

1/4*ln(tan(x))^2

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maxima [B]  time = 0.46, size = 265, normalized size = 29.44 \[ \frac {1}{4} \, {\left (\pi - 2 \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - 2 \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{4} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right )^{2} - \frac {1}{4} \, {\left (\pi - 2 \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + \frac {1}{4} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )^{2} - \frac {1}{4} \, \pi \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right ) + \frac {1}{4} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )^{2} + \frac {1}{8} \, {\left (\log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - \frac {1}{16} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )^{2} - \frac {1}{16} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )^{2} - \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - \frac {1}{16} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )^{2} - \frac {1}{2} \, \log \left (\cot \left (2 \, x\right ) + \csc \left (2 \, x\right )\right ) \log \left (\tan \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*log(tan(x)),x, algorithm="maxima")

[Out]

1/4*(pi - 2*arctan2(sin(x), cos(x) + 1) - 2*arctan2(sin(x), cos(x) - 1))*arctan2(sin(2*x), cos(2*x) + 1) + 1/4
*arctan2(sin(2*x), cos(2*x) + 1)^2 - 1/4*(pi - 2*arctan2(sin(x), cos(x) - 1))*arctan2(sin(x), cos(x) + 1) + 1/
4*arctan2(sin(x), cos(x) + 1)^2 - 1/4*pi*arctan2(sin(x), cos(x) - 1) + 1/4*arctan2(sin(x), cos(x) - 1)^2 + 1/8
*(log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1))*log(cos(2*x)^2 + sin(2*x)
^2 + 2*cos(2*x) + 1) - 1/16*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^2 - 1/16*log(cos(x)^2 + sin(x)^2 + 2
*cos(x) + 1)^2 - 1/8*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) - 1/16*lo
g(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)^2 - 1/2*log(cot(2*x) + csc(2*x))*log(tan(x))

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mupad [B]  time = 3.53, size = 27, normalized size = 3.00 \[ \frac {{\ln \left (-\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}+1}\right )}^2}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(tan(x))/sin(2*x),x)

[Out]

log(-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1))^2/4

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*x)*ln(tan(x)),x)

[Out]

Timed out

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