Optimal. Leaf size=115 \[ -\frac {3 i d^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac {3 i d (c+d x)^2}{2 b^2} \]
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Rubi [A] time = 0.17, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4410, 4184, 3717, 2190, 2279, 2391} \[ -\frac {3 i d^3 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac {3 i d (c+d x)^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4184
Rule 4410
Rubi steps
\begin {align*} \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx &=-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {(3 d) \int (c+d x)^2 \csc ^2(a+b x) \, dx}{2 b}\\ &=-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {\left (3 d^2\right ) \int (c+d x) \cot (a+b x) \, dx}{b^2}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac {\left (6 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)}{1-e^{2 i (a+b x)}} \, dx}{b^2}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {\left (3 d^3\right ) \int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {\left (3 i d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{2 b^4}\\ \end {align*}
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Mathematica [B] time = 6.41, size = 277, normalized size = 2.41 \[ \frac {3 c d^2 \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b^3 \left (\sin ^2(a)+\cos ^2(a)\right )}+\frac {3 \csc (a) \csc (a+b x) \left (c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right )}{2 b^2}-\frac {3 d^3 \csc (a) \sec (a) \left (b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac {\tan (a) \left (i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+i b x \left (2 \tan ^{-1}(\tan (a))-\pi \right )-2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )+2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\tan ^2(a)+1}}\right )}{2 b^4 \sqrt {\sec ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.75, size = 587, normalized size = 5.10 \[ \frac {b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + {\left (-3 i \, d^{3} \cos \left (b x + a\right )^{2} + 3 i \, d^{3}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (3 i \, d^{3} \cos \left (b x + a\right )^{2} - 3 i \, d^{3}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (3 i \, d^{3} \cos \left (b x + a\right )^{2} - 3 i \, d^{3}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (-3 i \, d^{3} \cos \left (b x + a\right )^{2} + 3 i \, d^{3}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (b d^{3} x + b c d^{2} - {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + b c d^{2} - {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b c d^{2} - a d^{3} - {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, {\left (b c d^{2} - a d^{3} - {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, {\left (b d^{3} x + a d^{3} - {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3} - {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right )}{2 \, {\left (b^{4} \cos \left (b x + a\right )^{2} - b^{4}\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 \[ \int {\left (d x + c\right )}^{3} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 409, normalized size = 3.56 \[ \frac {2 b \,d^{3} x^{3} {\mathrm e}^{2 i \left (b x +a \right )}-3 i d^{3} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+6 b c \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}-6 i c \,d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}+6 b \,c^{2} d x \,{\mathrm e}^{2 i \left (b x +a \right )}-3 i c^{2} d \,{\mathrm e}^{2 i \left (b x +a \right )}+3 i d^{3} x^{2}+2 b \,c^{3} {\mathrm e}^{2 i \left (b x +a \right )}+6 i c \,d^{2} x +3 i c^{2} d}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {3 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {3 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}-\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {3 i d^{3} x^{2}}{b^{2}}-\frac {6 i d^{3} a x}{b^{3}}-\frac {3 i d^{3} a^{2}}{b^{4}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{3}}-\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{4}}-\frac {3 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {3 d^{3} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}+\frac {6 d^{3} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 1044, normalized size = 9.08 \[ \text {result too large to display} \]
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 \[ \int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{{\sin \left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \left (c + d x\right )^{3} \cos {\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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