\(\int \log (a \csc ^n(x)) \, dx\) [178]

Optimal result

Integrand size = 7, antiderivative size = 51 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=-\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]

[Out]

-1/2*I*n*x^2+n*x*ln(1-exp(2*I*x))+x*ln(a*csc(x)^n)-1/2*I*n*polylog(2,exp(2*I*x))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3798, 2221, 2317, 2438} \[ \int \log \left (a \csc ^n(x)\right ) \, dx=x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right ) \]

[In]

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

[Out]

(-1/2*I)*n*x^2 + n*x*Log[1 - E^((2*I)*x)] + x*Log[a*Csc[x]^n] - (I/2)*n*PolyLog[2, E^((2*I)*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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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]

Rule 2628

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

Rule 3798

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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=-\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \ln \left (a \left (\csc ^{n}\left (x \right )\right )\right )d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (36) = 72\).

Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.29 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=n x \log \left (\frac {1}{\sin \left (x\right )}\right ) + \frac {1}{2} \, n x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + x \log \left (a\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \log \left (a \csc ^n(x)\right ) \, dx=\int \log {\left (a \csc ^{n}{\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

Integral(log(a*csc(x)**n), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (36) = 72\).

Time = 0.46 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.78 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=\frac {1}{2} \, {\left (-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 2 i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right )\right )} n + x \log \left (a \csc \left (x\right )^{n}\right ) \]

[In]

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

[Out]

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

Giac [F]

\[ \int \log \left (a \csc ^n(x)\right ) \, dx=\int { \log \left (a \csc \left (x\right )^{n}\right ) \,d x } \]

[In]

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

[Out]

integrate(log(a*csc(x)^n), x)

Mupad [F(-1)]

Timed out. \[ \int \log \left (a \csc ^n(x)\right ) \, dx=\int \ln \left (a\,{\left (\frac {1}{\sin \left (x\right )}\right )}^n\right ) \,d x \]

[In]

int(log(a*(1/sin(x))^n),x)

[Out]

int(log(a*(1/sin(x))^n), x)