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

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

[Out]

Log[Tan[x]]^2/4

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Rubi [A]  time = 0.0195538, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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*}

Mathematica [A]  time = 0.0109094, size = 9, normalized size = 1. \[ \frac{1}{4} \log ^2(\tan (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Tan[x]]^2/4

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Maple [A]  time = 0.016, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \tan \left ( x \right ) \right ) \right ) ^{2}}{4}} \end{align*}

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 = 1.54833, size = 358, normalized size = 39.78 \begin{align*} \frac{1}{4} \,{\left (\pi - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 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 \left (x\right ), \cos \left (x\right ) - 1\right )\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )^{2} - \frac{1}{4} \, \pi \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + \frac{1}{4} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )^{2} + \frac{1}{8} \,{\left (\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 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 \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )^{2} - \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{16} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )^{2} - \frac{1}{2} \, \log \left (\cot \left (2 \, x\right ) + \csc \left (2 \, x\right )\right ) \log \left (\tan \left (x\right )\right ) \end{align*}

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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Fricas [A]  time = 2.01379, size = 26, normalized size = 2.89 \begin{align*} \frac{1}{4} \, \log \left (\tan \left (x\right )\right )^{2} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (2 \, x\right ) \log \left (\tan \left (x\right )\right )\,{d x} \end{align*}

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)