Optimal. Leaf size=109 \[ \frac {i d \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x) \cot ^2(a+b x)}{2 b}-\frac {d x}{2 b}+\frac {i (c+d x)^2}{2 d} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3720, 3473, 8, 3717, 2190, 2279, 2391} \[ \frac {i d \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x) \cot ^2(a+b x)}{2 b}-\frac {d x}{2 b}+\frac {i (c+d x)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3473
Rule 3717
Rule 3720
Rubi steps
\begin {align*} \int (c+d x) \cot ^3(a+b x) \, dx &=-\frac {(c+d x) \cot ^2(a+b x)}{2 b}+\frac {d \int \cot ^2(a+b x) \, dx}{2 b}-\int (c+d x) \cot (a+b x) \, dx\\ &=\frac {i (c+d x)^2}{2 d}-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \cot ^2(a+b x)}{2 b}+2 i \int \frac {e^{2 i (a+b x)} (c+d x)}{1-e^{2 i (a+b x)}} \, dx-\frac {d \int 1 \, dx}{2 b}\\ &=-\frac {d x}{2 b}+\frac {i (c+d x)^2}{2 d}-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \cot ^2(a+b x)}{2 b}-\frac {(c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {d \int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {d x}{2 b}+\frac {i (c+d x)^2}{2 d}-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \cot ^2(a+b x)}{2 b}-\frac {(c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(i d) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^2}\\ &=-\frac {d x}{2 b}+\frac {i (c+d x)^2}{2 d}-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \cot ^2(a+b x)}{2 b}-\frac {(c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {i d \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^2}\\ \end {align*}
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Mathematica [B] time = 6.17, size = 240, normalized size = 2.20 \[ \frac {d \csc (a) \sec (a) \left (b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac {\tan (a) \left (i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+i b x \left (2 \tan ^{-1}(\tan (a))-\pi \right )-2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )+2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\tan ^2(a)+1}}\right )}{2 b^2 \sqrt {\sec ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}+\frac {d \csc (a) \sin (b x) \csc (a+b x)}{2 b^2}-\frac {c \left (\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))\right )}{2 b}-\frac {d x \csc ^2(a+b x)}{2 b}-\frac {1}{2} d x^2 \cot (a) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.46, size = 339, normalized size = 3.11 \[ \frac {4 \, b d x + 4 \, b c + {\left (i \, d \cos \left (2 \, b x + 2 \, a\right ) - i \, d\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + {\left (-i \, d \cos \left (2 \, b x + 2 \, a\right ) + i \, d\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \, {\left (b c - a d - {\left (b c - a d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b c - a d - {\left (b c - a d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, d \sin \left (2 \, b x + 2 \, a\right )}{4 \, {\left (b^{2} \cos \left (2 \, b x + 2 \, a\right ) - b^{2}\right )}} \]
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 \[ \int {\left (d x + c\right )} \cot \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 281, normalized size = 2.58 \[ \frac {i d \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-i c x +\frac {2 b d x \,{\mathrm e}^{2 i \left (b x +a \right )}+2 b c \,{\mathrm e}^{2 i \left (b x +a \right )}-i d \,{\mathrm e}^{2 i \left (b x +a \right )}+i d}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {c \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b}-\frac {c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b}+\frac {2 c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}+\frac {i d \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {i d \,x^{2}}{2}+\frac {i d \,a^{2}}{b^{2}}-\frac {d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}+\frac {2 i d a x}{b}-\frac {d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}-\frac {2 d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 839, normalized size = 7.70 \[ \text {result too large to display} \]
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 \[ \int {\mathrm {cot}\left (a+b\,x\right )}^3\,\left (c+d\,x\right ) \,d x \]
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 \[ \int \left (c + d x\right ) \cot ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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