Integrand size = 7, antiderivative size = 79 \[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=\frac {1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right )+i \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \sqrt {x}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \sqrt {x}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {2628, 12, 3833, 3798, 2221, 2611, 2320, 6724} \[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=i \sqrt {x} \operatorname {PolyLog}\left (2,e^{2 i \sqrt {x}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \sqrt {x}}\right )+\frac {1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right ) \]
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 2628
Rule 3798
Rule 3833
Rule 6724
Rubi steps \begin{align*} \text {integral}& = x \log \left (\sin \left (\sqrt {x}\right )\right )-\int \frac {1}{2} \sqrt {x} \cot \left (\sqrt {x}\right ) \, dx \\ & = x \log \left (\sin \left (\sqrt {x}\right )\right )-\frac {1}{2} \int \sqrt {x} \cot \left (\sqrt {x}\right ) \, dx \\ & = x \log \left (\sin \left (\sqrt {x}\right )\right )-\text {Subst}\left (\int x^2 \cot (x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{3} i x^{3/2}+x \log \left (\sin \left (\sqrt {x}\right )\right )+2 i \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right )+2 \text {Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right )+i \sqrt {x} \text {Li}_2\left (e^{2 i \sqrt {x}}\right )-i \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right )+i \sqrt {x} \text {Li}_2\left (e^{2 i \sqrt {x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sqrt {x}}\right ) \\ & = \frac {1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right )+i \sqrt {x} \text {Li}_2\left (e^{2 i \sqrt {x}}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 i \sqrt {x}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.11 \[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=\frac {i \pi ^3}{24}-\frac {1}{3} i x^{3/2}-x \log \left (1-e^{-2 i \sqrt {x}}\right )+x \log \left (\sin \left (\sqrt {x}\right )\right )-i \sqrt {x} \operatorname {PolyLog}\left (2,e^{-2 i \sqrt {x}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \sqrt {x}}\right ) \]
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\[\int \ln \left (\sin \left (\sqrt {x}\right )\right )d x\]
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\[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=\int { \log \left (\sin \left (\sqrt {x}\right )\right ) \,d x } \]
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\[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=\int \log {\left (\sin {\left (\sqrt {x} \right )} \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (49) = 98\).
Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.76 \[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=-i \, x \arctan \left (\sin \left (\sqrt {x}\right ), \cos \left (\sqrt {x}\right ) + 1\right ) + i \, x \arctan \left (\sin \left (\sqrt {x}\right ), -\cos \left (\sqrt {x}\right ) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \left (\sqrt {x}\right )^{2} + \sin \left (\sqrt {x}\right )^{2} + 2 \, \cos \left (\sqrt {x}\right ) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \left (\sqrt {x}\right )^{2} + \sin \left (\sqrt {x}\right )^{2} - 2 \, \cos \left (\sqrt {x}\right ) + 1\right ) + x \log \left (\sin \left (\sqrt {x}\right )\right ) + \frac {1}{3} i \, x^{\frac {3}{2}} + 2 i \, \sqrt {x} {\rm Li}_2\left (-e^{\left (i \, \sqrt {x}\right )}\right ) + 2 i \, \sqrt {x} {\rm Li}_2\left (e^{\left (i \, \sqrt {x}\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (i \, \sqrt {x}\right )}) - 2 \, {\rm Li}_{3}(e^{\left (i \, \sqrt {x}\right )}) \]
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\[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=\int { \log \left (\sin \left (\sqrt {x}\right )\right ) \,d x } \]
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Timed out. \[ \int \log \left (\sin \left (\sqrt {x}\right )\right ) \, dx=\int \ln \left (\sin \left (\sqrt {x}\right )\right ) \,d x \]
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