Optimal. Leaf size=74 \[ -\frac {i \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {2 x \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {x^2 \cot (a+b x)}{b}-\frac {i x^2}{b}-\frac {x^3}{3} \]
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Rubi [A] time = 0.12, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3720, 3717, 2190, 2279, 2391, 30} \[ -\frac {i \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {2 x \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {x^2 \cot (a+b x)}{b}-\frac {i x^2}{b}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 3720
Rubi steps
\begin {align*} \int x^2 \cot ^2(a+b x) \, dx &=-\frac {x^2 \cot (a+b x)}{b}+\frac {2 \int x \cot (a+b x) \, dx}{b}-\int x^2 \, dx\\ &=-\frac {i x^2}{b}-\frac {x^3}{3}-\frac {x^2 \cot (a+b x)}{b}-\frac {(4 i) \int \frac {e^{2 i (a+b x)} x}{1-e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac {i x^2}{b}-\frac {x^3}{3}-\frac {x^2 \cot (a+b x)}{b}+\frac {2 x \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 \int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i x^2}{b}-\frac {x^3}{3}-\frac {x^2 \cot (a+b x)}{b}+\frac {2 x \log \left (1-e^{2 i (a+b x)}\right )}{b^2}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^3}\\ &=-\frac {i x^2}{b}-\frac {x^3}{3}-\frac {x^2 \cot (a+b x)}{b}+\frac {2 x \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {i \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [B] time = 5.35, size = 153, normalized size = 2.07 \[ \frac {-b^2 x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt {\sec ^2(a)}-i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+i b x \left (\pi -2 \tan ^{-1}(\tan (a))\right )+2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )-2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )+\pi \log \left (1+e^{-2 i b x}\right )-\pi \log (\cos (b x))}{b^3}+\frac {x^2 \csc (a) \sin (b x) \csc (a+b x)}{b}-\frac {x^3}{3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.16, size = 281, normalized size = 3.80 \[ -\frac {2 \, b^{3} x^{3} \sin \left (2 \, b x + 2 \, a\right ) + 6 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + 6 \, b^{2} x^{2} + 6 \, a \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) + 6 \, a \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) + 3 i \, {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 i \, {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right )}{6 \, b^{3} \sin \left (2 \, b x + 2 \, a\right )} \]
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 \[ \int x^{2} \cot \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.90, size = 183, normalized size = 2.47 \[ -\frac {x^{3}}{3}-\frac {2 i x^{2}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {2 i x^{2}}{b}-\frac {4 i a x}{b^{2}}-\frac {2 i a^{2}}{b^{3}}+\frac {2 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}-\frac {2 i \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}-\frac {2 i \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {4 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 384, normalized size = 5.19 \[ \frac {-i \, b^{3} x^{3} + 6 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) + i \, b x \sin \left (2 \, b x + 2 \, a\right ) - b x\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 6 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) + i \, b x \sin \left (2 \, b x + 2 \, a\right ) - b x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) + {\left (i \, b^{3} x^{3} - 6 \, b^{2} x^{2}\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) - 1\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - 6 \, {\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) - 1\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + {\left (-3 i \, b x \cos \left (2 \, b x + 2 \, a\right ) + 3 \, b x \sin \left (2 \, b x + 2 \, a\right ) + 3 i \, b x\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + {\left (-3 i \, b x \cos \left (2 \, b x + 2 \, a\right ) + 3 \, b x \sin \left (2 \, b x + 2 \, a\right ) + 3 i \, b x\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - {\left (b^{3} x^{3} + 6 i \, b^{2} x^{2}\right )} \sin \left (2 \, b x + 2 \, a\right )}{-3 i \, b^{3} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, b^{3} \sin \left (2 \, b x + 2 \, a\right ) + 3 i \, b^{3}} \]
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 \[ \int x^2\,{\mathrm {cot}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{2} \cot ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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