Integrand size = 6, antiderivative size = 15 \[ \int \log (\sin (x)) \sin (x) \, dx=-\text {arctanh}(\cos (x))+\cos (x)-\cos (x) \log (\sin (x)) \]
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \log (\sin (x)) \sin (x) \, dx=\cos (x)-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )-\cos (x) \log (\sin (x)) \]
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3034, 25, 3042, 25, 3072, 262, 219}
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.
\(\displaystyle \int \sin (x) \log (\sin (x)) \, dx\) |
\(\Big \downarrow \) 3034 |
\(\displaystyle -\int -\cos (x) \cot (x)dx-\cos (x) \log (\sin (x))\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \cos (x) \cot (x)dx-\cos (x) \log (\sin (x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\sin \left (x+\frac {\pi }{2}\right ) \tan \left (x+\frac {\pi }{2}\right )dx-\cos (x) \log (\sin (x))\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sin \left (x+\frac {\pi }{2}\right ) \tan \left (x+\frac {\pi }{2}\right )dx-\cos (x) \log (\sin (x))\) |
\(\Big \downarrow \) 3072 |
\(\displaystyle -\int \frac {\cos ^2(x)}{1-\cos ^2(x)}d\cos (x)-\cos (x) \log (\sin (x))\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\int \frac {1}{1-\cos ^2(x)}d\cos (x)+\cos (x)+\cos (x) (-\log (\sin (x)))\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\text {arctanh}(\cos (x))+\cos (x)-\cos (x) \log (\sin (x))\) |
3.1.93.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Simp[Log[u] w, x ] - Int[SimplifyIntegrand[w*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(-\cos \left (x \right ) \ln \left (\sin \left (x \right )\right )+\cos \left (x \right )+\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )+1\) | \(20\) |
norman | \(\frac {2 \tan \left (\frac {x}{2}\right )^{2} \ln \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )+2}{1+\tan \left (\frac {x}{2}\right )^{2}}+\ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )\) | \(49\) |
default | \(-\frac {{\mathrm e}^{i x} \ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )}{2}+\frac {{\mathrm e}^{i x}}{2}+\ln \left (-1+{\mathrm e}^{i x}\right )-\ln \left ({\mathrm e}^{i x}+1\right )-\frac {{\mathrm e}^{-i x} \ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )}{2}+\frac {{\mathrm e}^{-i x}}{2}+\frac {\ln \left (2\right ) \left ({\mathrm e}^{i x}+{\mathrm e}^{-i x}\right )}{2}\) | \(111\) |
risch | \(\ln \left ({\mathrm e}^{i x}\right ) \cos \left (x \right )+\frac {{\mathrm e}^{-i x} \ln \left (2\right )}{2}-\frac {{\mathrm e}^{-i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}-\frac {{\mathrm e}^{i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}+\frac {{\mathrm e}^{i x} \ln \left (2\right )}{2}+\frac {{\mathrm e}^{i x}}{2}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (\sin \left (x \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \pi }{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (i \sin \left (x \right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) \pi }{4}+\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{-i x}}{2}-\ln \left ({\mathrm e}^{i x}+1\right )+\ln \left (-1+{\mathrm e}^{i x}\right )+\frac {i {\mathrm e}^{i x} \pi }{4}+\frac {i {\mathrm e}^{-i x} \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{4}-\frac {i {\mathrm e}^{-i x} \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{4}+\frac {i {\mathrm e}^{-i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}+\frac {i {\mathrm e}^{i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}-\frac {i {\mathrm e}^{i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2} \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{4}+\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2} \operatorname {csgn}\left (\sin \left (x \right )\right ) \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (\sin \left (x \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{4}\) | \(445\) |
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \log (\sin (x)) \sin (x) \, dx=-\cos \left (x\right ) \log \left (\sin \left (x\right )\right ) + \cos \left (x\right ) - \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \]
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (15) = 30\).
Time = 0.57 (sec) , antiderivative size = 105, normalized size of antiderivative = 7.00 \[ \int \log (\sin (x)) \sin (x) \, dx=\frac {2 \log {\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {2 \log {\left (2 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {2}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} \]
2*log(tan(x/2)/(tan(x/2)**2 + 1))*tan(x/2)**2/(tan(x/2)**2 + 1) + log(tan( x/2)**2 + 1)*tan(x/2)**2/(tan(x/2)**2 + 1) + log(tan(x/2)**2 + 1)/(tan(x/2 )**2 + 1) + 2*log(2)*tan(x/2)**2/(tan(x/2)**2 + 1) + 2/(tan(x/2)**2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (15) = 30\).
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 5.93 \[ \int \log (\sin (x)) \sin (x) \, dx=-\frac {2 \, \log \left (\frac {2 \, \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (x\right ) + 1\right )}}\right )}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} + \frac {2}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} - \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) + \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right ) \]
-2*log(2*sin(x)/((sin(x)^2/(cos(x) + 1)^2 + 1)*(cos(x) + 1)))/(sin(x)^2/(c os(x) + 1)^2 + 1) + 2/(sin(x)^2/(cos(x) + 1)^2 + 1) - log(sin(x)^2/(cos(x) + 1)^2 + 1) + log(sin(x)^2/(cos(x) + 1)^2)
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \log (\sin (x)) \sin (x) \, dx=-\cos \left (x\right ) \log \left (\sin \left (x\right )\right ) + \cos \left (x\right ) - \frac {1}{2} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (-\cos \left (x\right ) + 1\right ) \]
Timed out. \[ \int \log (\sin (x)) \sin (x) \, dx=\int \ln \left (\sin \left (x\right )\right )\,\sin \left (x\right ) \,d x \]