Optimal. Leaf size=74 \[ \frac {\text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}-\frac {i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^3}{3} \]
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Rubi [A] time = 0.14, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3717, 2190, 2531, 2282, 6589} \[ -\frac {i x \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {\text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^3}{3} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 6589
Rubi steps
\begin {align*} \int x^2 \cot (a+b x) \, dx &=-\frac {i x^3}{3}-2 i \int \frac {e^{2 i (a+b x)} x^2}{1-e^{2 i (a+b x)}} \, dx\\ &=-\frac {i x^3}{3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {2 \int x \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {i x^3}{3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {i \int \text {Li}_2\left (e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i x^3}{3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^3}\\ &=-\frac {i x^3}{3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {\text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 136, normalized size = 1.84 \[ \frac {3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \text {Li}_2\left (-e^{-i (a+b x)}\right )+6 i b x \text {Li}_2\left (e^{-i (a+b x)}\right )+6 \text {Li}_3\left (-e^{-i (a+b x)}\right )+6 \text {Li}_3\left (e^{-i (a+b x)}\right )+i b^3 x^3}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.57, size = 244, normalized size = 3.30 \[ \frac {-2 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cot \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.80, size = 198, normalized size = 2.68 \[ -\frac {i x^{3}}{3}+\frac {2 i a^{2} x}{b^{2}}-\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{3}}+\frac {4 i a^{3}}{3 b^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}-\frac {2 i \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {2 i \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 257, normalized size = 3.47 \[ -\frac {2 i \, {\left (b x + a\right )}^{3} - 6 i \, {\left (b x + a\right )}^{2} a + 12 i \, b x {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 12 i \, b x {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) - 6 \, a^{2} \log \left (\sin \left (b x + a\right )\right ) - {\left (6 i \, {\left (b x + a\right )}^{2} - 12 i \, {\left (b x + a\right )} a\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - {\left (-6 i \, {\left (b x + a\right )}^{2} + 12 i \, {\left (b x + a\right )} a\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 12 \, {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 12 \, {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )})}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {cot}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cot {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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