Optimal. Leaf size=45 \[ -i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \csc ^2(x)\right )-i x^2+2 x \log \left (1-e^{2 i x}\right ) \]
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Rubi [A] time = 0.057692, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {2548, 12, 3717, 2190, 2279, 2391} \[ -i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \csc ^2(x)\right )-i x^2+2 x \log \left (1-e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 12
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log \left (a \csc ^2(x)\right ) \, dx &=x \log \left (a \csc ^2(x)\right )-\int -2 x \cot (x) \, dx\\ &=x \log \left (a \csc ^2(x)\right )+2 \int x \cot (x) \, dx\\ &=-i x^2+x \log \left (a \csc ^2(x)\right )-4 i \int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx\\ &=-i x^2+2 x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^2(x)\right )-2 \int \log \left (1-e^{2 i 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{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-i x^2+2 x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^2(x)\right )-i \text{Li}_2\left (e^{2 i x}\right )\\ \end{align*}
Mathematica [A] time = 0.0171565, size = 42, normalized size = 0.93 \[ -i \left (x^2+\text{PolyLog}\left (2,e^{2 i x}\right )\right )+x \log \left (a \csc ^2(x)\right )+2 x \log \left (1-e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 88, normalized size = 2. \begin{align*} i \left ( \ln \left ({{\rm e}^{ix}} \right ) \right ) ^{2}-i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ( -{\frac{a{{\rm e}^{2\,ix}}}{ \left ({{\rm e}^{2\,ix}}-1 \right ) ^{2}}} \right ) -2\,i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ({{\rm e}^{ix}}+1 \right ) -2\,i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ( 2 \right ) -2\,i{\it dilog} \left ({{\rm e}^{ix}}+1 \right ) +2\,i{\it dilog} \left ({{\rm e}^{ix}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.95529, size = 117, normalized size = 2.6 \begin{align*} -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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01015, size = 365, normalized size = 8.11 \begin{align*} 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 ) \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 (a \csc ^{2}{\left (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 (a \csc \left (x\right )^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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