Integrand size = 10, antiderivative size = 101 \[ \int x^3 \cot (a+b x) \, dx=-\frac {i x^4}{4}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \]
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Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3798, 2221, 2611, 6744, 2320, 6724} \[ \int x^3 \cot (a+b x) \, dx=\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^4}{4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\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 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {(3 i) \int x \operatorname {PolyLog}\left (2,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 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 \int \operatorname {PolyLog}\left (3,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 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {(3 i) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.82 \[ \int x^3 \cot (a+b x) \, dx=\frac {i b^4 x^4+4 b^3 x^3 \log \left (1-e^{-i (a+b x)}\right )+4 b^3 x^3 \log \left (1+e^{-i (a+b x)}\right )+12 i b^2 x^2 \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+12 i b^2 x^2 \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+24 b x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+24 b x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )-24 i \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )-24 i \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )}{4 b^4} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (82 ) = 164\).
Time = 0.49 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.38
method | result | size |
risch | \(-\frac {i x^{4}}{4}-\frac {3 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {2 i a^{3} x}{b^{3}}+\frac {\ln \left (1+{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {a^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {3 i a^{4}}{2 b^{4}}+\frac {6 i \operatorname {polylog}\left (4, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 i \operatorname {polylog}\left (4, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}\) | \(240\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (78) = 156\).
Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int x^3 \cot (a+b x) \, dx=\frac {-6 i \, b^{2} x^{2} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 6 i \, b^{2} x^{2} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 4 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 4 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 6 \, b x {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 6 \, b x {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 4 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 4 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 i \, {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 3 i \, {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{8 \, b^{4}} \]
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\[ \int x^3 \cot (a+b x) \, dx=\int x^{3} \cot {\left (a + b x \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (78) = 156\).
Time = 0.36 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.87 \[ \int x^3 \cot (a+b x) \, dx=-\frac {i \, {\left (b x + a\right )}^{4} - 4 i \, {\left (b x + a\right )}^{3} a + 6 i \, {\left (b x + a\right )}^{2} a^{2} + 4 \, a^{3} \log \left (\sin \left (b x + a\right )\right ) - 24 \, b x {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 24 \, b x {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )}) + 4 \, {\left (-i \, {\left (b x + a\right )}^{3} + 3 i \, {\left (b x + a\right )}^{2} a - 3 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) + 4 \, {\left (i \, {\left (b x + a\right )}^{3} - 3 i \, {\left (b x + a\right )}^{2} a + 3 i \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) + 12 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a + i \, a^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 12 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a + i \, a^{2}\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) - 2 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 2 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 24 i \, {\rm Li}_{4}(-e^{\left (i \, b x + i \, a\right )}) - 24 i \, {\rm Li}_{4}(e^{\left (i \, b x + i \, a\right )})}{4 \, b^{4}} \]
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\[ \int x^3 \cot (a+b x) \, dx=\int { x^{3} \cot \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^3 \cot (a+b x) \, dx=\int x^3\,\mathrm {cot}\left (a+b\,x\right ) \,d x \]
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