Optimal. Leaf size=254 \[ -\frac {6 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.20, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4408, 3296, 2637, 4183, 2531, 6609, 2282, 6589} \[ -\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {6 i d^3 \text {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \text {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {6 d^3 \sin (a+b x)}{b^4}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 4183
Rule 4408
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int (c+d x)^3 \cos (a+b x) \cot (a+b x) \, dx &=\int (c+d x)^3 \csc (a+b x) \, dx-\int (c+d x)^3 \sin (a+b x) \, dx\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {(3 d) \int (c+d x)^2 \cos (a+b x) \, dx}{b}-\frac {(3 d) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {\left (6 i d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x) \text {Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{b^2}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {\left (6 d^3\right ) \int \cos (a+b x) \, dx}{b^3}+\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (6 d^3\right ) \int \text {Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {\left (6 i d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (6 i d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 330, normalized size = 1.30 \[ \frac {-2 b^3 (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))+3 i d \left (b^2 (c+d x)^2 \text {Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text {Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text {Li}_4(-\cos (a+b x)-i \sin (a+b x))\right )-3 i d \left (b^2 (c+d x)^2 \text {Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text {Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text {Li}_4(\cos (a+b x)+i \sin (a+b x))\right )+\cos (b x) \left (b \cos (a) (c+d x) \left (b^2 (c+d x)^2-6 d^2\right )-3 d \sin (a) \left (b^2 (c+d x)^2-2 d^2\right )\right )-\sin (b x) \left (b \sin (a) (c+d x) \left (b^2 (c+d x)^2-6 d^2\right )+3 d \cos (a) \left (b^2 (c+d x)^2-2 d^2\right )\right )}{b^4} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.79, size = 921, normalized size = 3.63 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{3} \cos \left (b x + a\right ) \cot \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 847, normalized size = 3.33 \[ -\frac {6 i c \,d^{2} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {6 i c \,d^{2} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{3}}{b^{4}}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{3}}{b}+\frac {6 i d^{3} \polylog \left (4, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {6 c \,d^{2} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 c \,d^{2} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 i d^{3} \polylog \left (4, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 i b^{2} d^{3} x^{2}+3 b^{3} c^{2} d x +6 i b^{2} c \,d^{2} x +b^{3} c^{3}+3 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b -6 i d^{3}\right ) {\mathrm e}^{i \left (b x +a \right )}}{2 b^{4}}+\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}-3 i b^{2} d^{3} x^{2}+3 b^{3} c^{2} d x -6 i b^{2} c \,d^{2} x +b^{3} c^{3}-3 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b +6 i d^{3}\right ) {\mathrm e}^{-i \left (b x +a \right )}}{2 b^{4}}-\frac {2 c^{3} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {3 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{2}}+\frac {3 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}-\frac {6 c \,d^{2} a^{2} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {6 c^{2} d a \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 i c^{2} d \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 i c^{2} d \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {3 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {3 c \,d^{2} a^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a^{3}}{b^{4}}+\frac {2 d^{3} a^{3} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 919, normalized size = 3.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \cos \left (a+b\,x\right )\,\mathrm {cot}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^3 \,d x \]
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 \[ \int \left (c + d x\right )^{3} \cos {\left (a + b x \right )} \cot {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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