Optimal. Leaf size=79 \[ i \sqrt{x} \text{PolyLog}\left (2,e^{2 i \sqrt{x}}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 i \sqrt{x}}\right )+\frac{1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt{x}}\right )+x \log \left (\sin \left (\sqrt{x}\right )\right ) \]
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Rubi [A] time = 0.102098, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.143, Rules used = {2548, 12, 3748, 3717, 2190, 2531, 2282, 6589} \[ i \sqrt{x} \text{PolyLog}\left (2,e^{2 i \sqrt{x}}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 i \sqrt{x}}\right )+\frac{1}{3} i x^{3/2}-x \log \left (1-e^{2 i \sqrt{x}}\right )+x \log \left (\sin \left (\sqrt{x}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 12
Rule 3748
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \log \left (\sin \left (\sqrt{x}\right )\right ) \, dx &=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 )-\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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} \operatorname{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*}
Mathematica [A] time = 0.0324164, size = 88, normalized size = 1.11 \[ -i \sqrt{x} \text{PolyLog}\left (2,e^{-2 i \sqrt{x}}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{-2 i \sqrt{x}}\right )-\frac{1}{3} i x^{3/2}-x \log \left (1-e^{-2 i \sqrt{x}}\right )+x \log \left (\sin \left (\sqrt{x}\right )\right )+\frac{i \pi ^3}{24} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( \sin \left ( \sqrt{x} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06539, size = 188, normalized size = 2.38 \begin{align*} -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 )}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left (\sin \left (\sqrt{x}\right )\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\sin{\left (\sqrt{x} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (\sin \left (\sqrt{x}\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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