Optimal. Leaf size=97 \[ \frac {3 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {x^3 \cot (a+b x)}{b}-\frac {i x^3}{b}-\frac {x^4}{4} \]
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Rubi [A] time = 0.18, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3720, 3717, 2190, 2531, 2282, 6589, 30} \[ -\frac {3 i x \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {x^3 \cot (a+b x)}{b}-\frac {i x^3}{b}-\frac {x^4}{4} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 3720
Rule 6589
Rubi steps
\begin {align*} \int x^3 \cot ^2(a+b x) \, dx &=-\frac {x^3 \cot (a+b x)}{b}+\frac {3 \int x^2 \cot (a+b x) \, dx}{b}-\int x^3 \, dx\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}-\frac {x^3 \cot (a+b x)}{b}-\frac {(6 i) \int \frac {e^{2 i (a+b x)} x^2}{1-e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}-\frac {x^3 \cot (a+b x)}{b}+\frac {3 x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {6 \int x \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}-\frac {x^3 \cot (a+b x)}{b}+\frac {3 x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {(3 i) \int \text {Li}_2\left (e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}-\frac {x^3 \cot (a+b x)}{b}+\frac {3 x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac {i x^3}{b}-\frac {x^4}{4}-\frac {x^3 \cot (a+b x)}{b}+\frac {3 x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 171, normalized size = 1.76 \[ \frac {-\frac {2 i b^3 x^3}{-1+e^{2 i a}}+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \text {Li}_2\left (-e^{-i (a+b x)}\right )+6 i b x \text {Li}_2\left (e^{-i (a+b x)}\right )+6 \text {Li}_3\left (-e^{-i (a+b x)}\right )+6 \text {Li}_3\left (e^{-i (a+b x)}\right )}{b^4}+\frac {x^3 \csc (a) \sin (b x) \csc (a+b x)}{b}-\frac {x^4}{4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.69, size = 372, normalized size = 3.84 \[ -\frac {b^{4} x^{4} \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{3} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + 4 \, b^{3} x^{3} + 6 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right )}{4 \, b^{4} \sin \left (2 \, b x + 2 \, a\right )} \]
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 \[ \int x^{3} \cot \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.95, size = 231, normalized size = 2.38 \[ -\frac {x^{4}}{4}-\frac {2 i x^{3}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {6 i a^{2} x}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}-\frac {6 i \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{4}}-\frac {6 i \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {2 i x^{3}}{b}+\frac {4 i a^{3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 953, normalized size = 9.82 \[ \text {result too large to display} \]
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 \[ \int x^3\,{\mathrm {cot}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{3} \cot ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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