Optimal. Leaf size=53 \[ -\frac {i \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}+\frac {x \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^2}{2} \]
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Rubi [A] time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3717, 2190, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {x \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rubi steps
\begin {align*} \int x \cot (a+b x) \, dx &=-\frac {i x^2}{2}-2 i \int \frac {e^{2 i (a+b x)} x}{1-e^{2 i (a+b x)}} \, dx\\ &=-\frac {i x^2}{2}+\frac {x \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {\int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {i x^2}{2}+\frac {x \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^2}\\ &=-\frac {i x^2}{2}+\frac {x \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}\\ \end {align*}
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Mathematica [B] time = 3.73, size = 135, normalized size = 2.55 \[ \frac {1}{2} \left (-\frac {i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )-i b x \left (\pi -2 \tan ^{-1}(\tan (a))\right )-2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )+2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))}{b^2}+x^2 \cot (a)-x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt {\sec ^2(a)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.53, size = 174, normalized size = 3.28 \[ -\frac {2 \, a \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, a \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 2 \, {\left (b x + a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) - 2 \, {\left (b x + a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + i \, {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - i \, {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{4 \, b^{2}} \]
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 \[ \int x \cot \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.82, size = 150, normalized size = 2.83 \[ -\frac {i x^{2}}{2}-\frac {2 i a x}{b}-\frac {i a^{2}}{b^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}-\frac {i \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {i \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 140, normalized size = 2.64 \[ \frac {-i \, b^{2} x^{2} + 2 i \, b x \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 2 i \, b x \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) + b x \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + b x \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - 2 i \, {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {cot}\left (a+b\,x\right ) \,d x \]
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 \[ \int x \cot {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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