Optimal. Leaf size=52 \[ \frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text {li}(x)+\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2629, 3798,
2221, 2317, 2438, 2335} \begin {gather*} \frac {1}{2} i \text {PolyLog}\left (2,e^{2 i x}\right )-\text {li}(x)+\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2317
Rule 2335
Rule 2438
Rule 2629
Rule 3798
Rubi steps
\begin {align*} \int \log (\log (x) \sin (x)) \, dx &=x \log (\log (x) \sin (x))-\int \left (x \cot (x)+\frac {1}{\log (x)}\right ) \, dx\\ &=x \log (\log (x) \sin (x))-\int x \cot (x) \, dx-\int \frac {1}{\log (x)} \, dx\\ &=\frac {i x^2}{2}+x \log (\log (x) \sin (x))-\text {li}(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 (\log (x) \sin (x))-\text {li}(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 (\log (x) \sin (x))-\text {li}(x)-\frac {1}{2} i \text {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 (\log (x) \sin (x))-\text {li}(x)+\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 47, normalized size = 0.90 \begin {gather*} -x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text {li}(x)+\frac {1}{2} i \left (x^2+\text {Li}_2\left (e^{2 i x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 368, normalized size = 7.08
method | result | size |
risch | \(-x \ln \left ({\mathrm e}^{i x}\right )+i \dilog \left ({\mathrm e}^{i x}+1\right )+\frac {i x \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right ) \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2}}{2}-\frac {i x \pi \,\mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2}}{2}-\frac {i x \pi }{2}-\frac {i x \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )^{3}}{2}-x \ln \left (2\right )+\frac {i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2}}{2}-i \dilog \left ({\mathrm e}^{i x}\right )-\frac {i x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )}{2}+\frac {i x \pi \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3}}{2}+\frac {i x \pi \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2}}{2}-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )-\frac {i x \pi \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3}}{2}+\frac {i x^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right ) \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )}{2}+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )+\frac {i x \pi \,\mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )}{2}+\ln \left (\ln \left (x \right )\right ) x +\expIntegral \left (1, -\ln \left (x \right )\right )\) | \(368\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.61, size = 43, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, {\left (i \, \pi - 2 \, \log \left (2\right )\right )} x - \frac {1}{2} i \, x^{2} + x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 109 vs. \(2 (37) = 74\).
time = 0.44, size = 109, normalized size = 2.10 \begin {gather*} x \log \left (\log \left (x\right ) \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 ) - \operatorname {log\_integral}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \ln \left (\ln \left (x\right )\,\sin \left (x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________