Optimal. Leaf size=101 \[ \frac {3 i \text {Li}_4\left (e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 x \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^4}{4} \]
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Rubi [A] time = 0.17, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3717, 2190, 2531, 6609, 2282, 6589} \[ -\frac {3 i x^2 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \text {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^4}{4} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \cot (a+b x) \, dx &=-\frac {i x^4}{4}-2 i \int \frac {e^{2 i (a+b x)} x^3}{1-e^{2 i (a+b x)}} \, dx\\ &=-\frac {i x^4}{4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 \int x^2 \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {i x^4}{4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}+\frac {(3 i) \int x \text {Li}_2\left (e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i x^4}{4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 \int \text {Li}_3\left (e^{2 i (a+b x)}\right ) \, dx}{2 b^3}\\ &=-\frac {i x^4}{4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{4 b^4}\\ &=-\frac {i x^4}{4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \text {Li}_4\left (e^{2 i (a+b x)}\right )}{4 b^4}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 184, normalized size = 1.82 \[ \frac {4 b^3 x^3 \log \left (1-e^{-i (a+b x)}\right )+4 b^3 x^3 \log \left (1+e^{-i (a+b x)}\right )+12 i b^2 x^2 \text {Li}_2\left (-e^{-i (a+b x)}\right )+12 i b^2 x^2 \text {Li}_2\left (e^{-i (a+b x)}\right )+24 b x \text {Li}_3\left (-e^{-i (a+b x)}\right )+24 b x \text {Li}_3\left (e^{-i (a+b x)}\right )-24 i \text {Li}_4\left (-e^{-i (a+b x)}\right )-24 i \text {Li}_4\left (e^{-i (a+b x)}\right )+i b^4 x^4}{4 b^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.56, size = 306, normalized size = 3.03 \[ \frac {-6 i \, b^{2} x^{2} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 4 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 4 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 6 \, b x {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 6 \, b x {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 4 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 4 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 i \, {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 3 i \, {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{8 \, b^{4}} \]
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 \[ \int x^{3} \cot \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.86, size = 240, normalized size = 2.38 \[ -\frac {i x^{4}}{4}-\frac {3 i \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 i \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {2 i a^{3} x}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{3}}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{3}}{b}+\frac {6 i \polylog \left (4, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 i \polylog \left (4, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {3 i a^{4}}{2 b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}+\frac {6 \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 391, normalized size = 3.87 \[ -\frac {i \, {\left (b x + a\right )}^{4} - 4 i \, {\left (b x + a\right )}^{3} a + 6 i \, {\left (b x + a\right )}^{2} a^{2} + 4 \, a^{3} \log \left (\sin \left (b x + a\right )\right ) - 24 \, b x {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 24 \, b x {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )}) - {\left (4 i \, {\left (b x + a\right )}^{3} - 12 i \, {\left (b x + a\right )}^{2} a + 12 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - {\left (-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 12 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - {\left (-12 i \, {\left (b x + a\right )}^{2} + 24 i \, {\left (b x + a\right )} a - 12 i \, a^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - {\left (-12 i \, {\left (b x + a\right )}^{2} + 24 i \, {\left (b x + a\right )} a - 12 i \, a^{2}\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) - 2 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 2 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 24 i \, {\rm Li}_{4}(-e^{\left (i \, b x + i \, a\right )}) - 24 i \, {\rm Li}_{4}(e^{\left (i \, b x + i \, a\right )})}{4 \, b^{4}} \]
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 \[ \int x^3\,\mathrm {cot}\left (a+b\,x\right ) \,d x \]
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 \[ \int x^{3} \cot {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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