3.171 \(\int \log (a \cot ^2(x)) \, dx\)

Optimal. Leaf size=49 \[ i \text{PolyLog}\left (2,-e^{2 i x}\right )-i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \cot ^2(x)\right )-4 x \tanh ^{-1}\left (e^{2 i x}\right ) \]

[Out]

-4*x*ArcTanh[E^((2*I)*x)] + x*Log[a*Cot[x]^2] + I*PolyLog[2, -E^((2*I)*x)] - I*PolyLog[2, E^((2*I)*x)]

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Rubi [A]  time = 0.0471132, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {2548, 12, 4419, 4183, 2279, 2391} \[ i \text{PolyLog}\left (2,-e^{2 i x}\right )-i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \cot ^2(x)\right )-4 x \tanh ^{-1}\left (e^{2 i x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[a*Cot[x]^2],x]

[Out]

-4*x*ArcTanh[E^((2*I)*x)] + x*Log[a*Cot[x]^2] + I*PolyLog[2, -E^((2*I)*x)] - I*PolyLog[2, E^((2*I)*x)]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4419

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
 e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \log \left (a \cot ^2(x)\right ) \, dx &=x \log \left (a \cot ^2(x)\right )-\int -2 x \csc (x) \sec (x) \, dx\\ &=x \log \left (a \cot ^2(x)\right )+2 \int x \csc (x) \sec (x) \, dx\\ &=x \log \left (a \cot ^2(x)\right )+4 \int x \csc (2 x) \, dx\\ &=-4 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^2(x)\right )-2 \int \log \left (1-e^{2 i x}\right ) \, dx+2 \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=-4 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^2(x)\right )+i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )-i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-4 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^2(x)\right )+i \text{Li}_2\left (-e^{2 i x}\right )-i \text{Li}_2\left (e^{2 i x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0115815, size = 75, normalized size = 1.53 \[ i \text{PolyLog}(2,-i \tan (x))-i \text{PolyLog}(2,i \tan (x))-\frac{1}{2} i \log (-i (-\tan (x)+i)) \log \left (a \cot ^2(x)\right )+\frac{1}{2} i \log (-i (\tan (x)+i)) \log \left (a \cot ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a*Cot[x]^2],x]

[Out]

(-I/2)*Log[a*Cot[x]^2]*Log[(-I)*(I - Tan[x])] + (I/2)*Log[a*Cot[x]^2]*Log[(-I)*(I + Tan[x])] + I*PolyLog[2, (-
I)*Tan[x]] - I*PolyLog[2, I*Tan[x]]

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Maple [A]  time = 0.043, size = 82, normalized size = 1.7 \begin{align*}{\frac{i}{2}}\ln \left ( a \left ( \cot \left ( x \right ) \right ) ^{2} \right ) \ln \left ( \cot \left ( x \right ) -i \right ) -i\ln \left ( \cot \left ( x \right ) -i \right ) \ln \left ( -i\cot \left ( x \right ) \right ) -i{\it dilog} \left ( -i\cot \left ( x \right ) \right ) -{\frac{i}{2}}\ln \left ( a \left ( \cot \left ( x \right ) \right ) ^{2} \right ) \ln \left ( \cot \left ( x \right ) +i \right ) +i\ln \left ( \cot \left ( x \right ) +i \right ) \ln \left ( i\cot \left ( x \right ) \right ) +i{\it dilog} \left ( i\cot \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*cot(x)^2),x)

[Out]

1/2*I*ln(a*cot(x)^2)*ln(cot(x)-I)-I*ln(cot(x)-I)*ln(-I*cot(x))-I*dilog(-I*cot(x))-1/2*I*ln(a*cot(x)^2)*ln(cot(
x)+I)+I*ln(cot(x)+I)*ln(I*cot(x))+I*dilog(I*cot(x))

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Maxima [A]  time = 1.52934, size = 59, normalized size = 1.2 \begin{align*} -\frac{1}{2} \, \pi \log \left (\tan \left (x\right )^{2} + 1\right ) + x \log \left (\frac{a}{\tan \left (x\right )^{2}}\right ) + 2 \, x \log \left (\tan \left (x\right )\right ) - i \,{\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) + i \,{\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*cot(x)^2),x, algorithm="maxima")

[Out]

-1/2*pi*log(tan(x)^2 + 1) + x*log(a/tan(x)^2) + 2*x*log(tan(x)) - I*dilog(I*tan(x) + 1) + I*dilog(-I*tan(x) +
1)

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Fricas [B]  time = 2.13031, size = 450, normalized size = 9.18 \begin{align*} x \log \left (-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}\right ) - x \log \left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) - x \log \left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) + x \log \left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) + x \log \left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) - \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*cot(x)^2),x, algorithm="fricas")

[Out]

x*log(-(a*cos(2*x) + a)/(cos(2*x) - 1)) - x*log(cos(2*x) + I*sin(2*x) + 1) - x*log(cos(2*x) - I*sin(2*x) + 1)
+ x*log(-cos(2*x) + I*sin(2*x) + 1) + x*log(-cos(2*x) - I*sin(2*x) + 1) - 1/2*I*dilog(cos(2*x) + I*sin(2*x)) +
 1/2*I*dilog(cos(2*x) - I*sin(2*x)) - 1/2*I*dilog(-cos(2*x) + I*sin(2*x)) + 1/2*I*dilog(-cos(2*x) - I*sin(2*x)
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (a \cot ^{2}{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*cot(x)**2),x)

[Out]

Integral(log(a*cot(x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*cot(x)^2),x, algorithm="giac")

[Out]

integrate(log(a*cot(x)^2), x)