Integrand size = 7, antiderivative size = 45 \[ \int \log \left (a \csc ^2(x)\right ) \, dx=-i x^2+2 x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^2(x)\right )-i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \log \left (a \csc ^2(x)\right ) \, dx=2 x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^2(x)\right )-i \left (x^2+\operatorname {PolyLog}\left (2,e^{2 i x}\right )\right ) \]
Time = 0.32 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3028, 27, 3042, 25, 4200, 25, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \log \left (a \csc ^2(x)\right ) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-\int -2 x \cot (x)dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int x \cot (x)dx+x \log \left (a \csc ^2(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int -x \tan \left (x+\frac {\pi }{2}\right )dx+x \log \left (a \csc ^2(x)\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-2 \int x \tan \left (x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-2 \left (\frac {i x^2}{2}-2 i \int -\frac {e^{2 i x} x}{1-e^{2 i x}}dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-2 \left (2 i \int \frac {e^{2 i x} x}{1-e^{2 i x}}dx+\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-2 \left (2 i \left (\frac {1}{2} i x \log \left (1-e^{2 i x}\right )-\frac {1}{2} i \int \log \left (1-e^{2 i x}\right )dx\right )+\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-2 \left (2 i \left (\frac {1}{2} i x \log \left (1-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \log \left (1-e^{2 i x}\right )de^{2 i x}\right )+\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log \left (a \csc ^2(x)\right )-2 \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {1}{2} i x \log \left (1-e^{2 i x}\right )\right )+\frac {i x^2}{2}\right )\) |
x*Log[a*Csc[x]^2] - 2*((I/2)*x^2 + (2*I)*((I/2)*x*Log[1 - E^((2*I)*x)] + P olyLog[2, E^((2*I)*x)]/4))
3.2.77.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[u, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (39 ) = 78\).
Time = 1.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.96
method | result | size |
default | \(-i \left (\ln \left ({\mathrm e}^{i x}\right ) \ln \left (-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )-\ln \left ({\mathrm e}^{i x}\right )^{2}+2 \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )+2 \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )-2 \operatorname {dilog}\left ({\mathrm e}^{i x}\right )+2 \ln \left (2\right ) \ln \left ({\mathrm e}^{i x}\right )\right )\) | \(88\) |
risch | \(2 x \ln \left ({\mathrm e}^{i x}\right )+2 i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )-i x^{2}+\frac {i \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )^{2}\right )}^{3} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right )^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right ) x}{2}+i \pi x -2 i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{3} x}{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right )^{2} x +\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}+2 x \ln \left (2\right )+\ln \left (a \right ) x -2 i \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )+2 i \operatorname {dilog}\left ({\mathrm e}^{i x}\right )-i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{2} x +\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right )^{3} x}{2}-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )^{2}\right )}^{2} x +\frac {i \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )^{2}\right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{i x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}\) | \(549\) |
-I*(ln(exp(I*x))*ln(-a*exp(I*x)^2/(exp(I*x)^2-1)^2)-ln(exp(I*x))^2+2*ln(ex p(I*x))*ln(exp(I*x)+1)+2*dilog(exp(I*x)+1)-2*dilog(exp(I*x))+2*ln(2)*ln(ex p(I*x)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (34) = 68\).
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.38 \[ \int \log \left (a \csc ^2(x)\right ) \, dx=x \log \left (-\frac {a}{\cos \left (x\right )^{2} - 1}\right ) + x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - i \, {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + i \, {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + i \, {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - i \, {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \]
x*log(-a/(cos(x)^2 - 1)) + x*log(cos(x) + I*sin(x) + 1) + x*log(cos(x) - I *sin(x) + 1) + x*log(-cos(x) + I*sin(x) + 1) + x*log(-cos(x) - I*sin(x) + 1) - I*dilog(cos(x) + I*sin(x)) + I*dilog(cos(x) - I*sin(x)) + I*dilog(-co s(x) + I*sin(x)) - I*dilog(-cos(x) - I*sin(x))
\[ \int \log \left (a \csc ^2(x)\right ) \, dx=\int \log {\left (a \csc ^{2}{\left (x \right )} \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (34) = 68\).
Time = 0.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.93 \[ \int \log \left (a \csc ^2(x)\right ) \, dx=-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x \log \left (a \csc \left (x\right )^{2}\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 2 i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \]
-I*x^2 + 2*I*x*arctan2(sin(x), cos(x) + 1) - 2*I*x*arctan2(sin(x), -cos(x) + 1) + x*log(a*csc(x)^2) + 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))
\[ \int \log \left (a \csc ^2(x)\right ) \, dx=\int { \log \left (a \csc \left (x\right )^{2}\right ) \,d x } \]
Timed out. \[ \int \log \left (a \csc ^2(x)\right ) \, dx=\int \ln \left (\frac {a}{{\sin \left (x\right )}^2}\right ) \,d x \]