### 3.176 $$\int \log (a \csc (x)) \, dx$$

Optimal. Leaf size=46 $-\frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log (a \csc (x))-\frac{i x^2}{2}+x \log \left (1-e^{2 i x}\right )$

[Out]

(-I/2)*x^2 + x*Log[1 - E^((2*I)*x)] + x*Log[a*Csc[x]] - (I/2)*PolyLog[2, E^((2*I)*x)]

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Rubi [A]  time = 0.0528519, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 5, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 1., Rules used = {2548, 3717, 2190, 2279, 2391} $-\frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log (a \csc (x))-\frac{i x^2}{2}+x \log \left (1-e^{2 i x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[a*Csc[x]],x]

[Out]

(-I/2)*x^2 + x*Log[1 - E^((2*I)*x)] + x*Log[a*Csc[x]] - (I/2)*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 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, 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 (a \csc (x)) \, dx &=x \log (a \csc (x))+\int x \cot (x) \, dx\\ &=-\frac{i x^2}{2}+x \log (a \csc (x))-2 i \int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx\\ &=-\frac{i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))-\int \log \left (1-e^{2 i x}\right ) \, dx\\ &=-\frac{i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-\frac{i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))-\frac{1}{2} i \text{Li}_2\left (e^{2 i x}\right )\\ \end{align*}

Mathematica [A]  time = 0.012003, size = 41, normalized size = 0.89 $-\frac{1}{2} i \left (x^2+\text{PolyLog}\left (2,e^{2 i x}\right )\right )+x \log (a \csc (x))+x \log \left (1-e^{2 i x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[a*Csc[x]],x]

[Out]

x*Log[1 - E^((2*I)*x)] + x*Log[a*Csc[x]] - (I/2)*(x^2 + PolyLog[2, E^((2*I)*x)])

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Maple [B]  time = 0.033, size = 89, normalized size = 1.9 \begin{align*} -i\ln \left ( 2 \right ) \ln \left ({{\rm e}^{ix}} \right ) -i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ({\frac{ia{{\rm e}^{ix}}}{{{\rm e}^{2\,ix}}-1}} \right ) +{\frac{i}{2}} \left ( \ln \left ({{\rm e}^{ix}} \right ) \right ) ^{2}+i{\it dilog} \left ({{\rm e}^{ix}} \right ) -i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ({{\rm e}^{ix}}+1 \right ) -i{\it dilog} \left ({{\rm e}^{ix}}+1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*csc(x)),x)

[Out]

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

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Maxima [B]  time = 2.16309, size = 117, normalized size = 2.54 \begin{align*} -\frac{1}{2} i \, x^{2} + i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (a \csc \left (x\right )\right ) - i \,{\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - i \,{\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*x^2 + I*x*arctan2(sin(x), cos(x) + 1) - I*x*arctan2(sin(x), -cos(x) + 1) + 1/2*x*log(cos(x)^2 + sin(x)^
2 + 2*cos(x) + 1) + 1/2*x*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + x*log(a*csc(x)) - I*dilog(-e^(I*x)) - I*di
log(e^(I*x))

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Fricas [B]  time = 2.18976, size = 396, normalized size = 8.61 \begin{align*} x \log \left (\frac{a}{\sin \left (x\right )}\right ) + \frac{1}{2} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*csc(x)),x)

[Out]

Integral(log(a*csc(x)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*csc(x)), x)