Integrand size = 22, antiderivative size = 115 \[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4} \]
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Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4495, 4269, 3798, 2221, 2317, 2438} \[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac {3 i d (c+d x)^2}{2 b^2} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4269
Rule 4495
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(115)=230\).
Time = 6.46 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.41 \[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {3 c d^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {3 \csc (a) \csc (a+b x) \left (c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right )}{2 b^2}-\frac {3 d^3 \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{2 b^4 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (101 ) = 202\).
Time = 1.67 (sec) , antiderivative size = 409, normalized size of antiderivative = 3.56
method | result | size |
risch | \(\frac {2 b \,d^{3} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}-3 i d^{3} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}+6 b c \,d^{2} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}-6 i c \,d^{2} x \,{\mathrm e}^{2 i \left (x b +a \right )}+6 b \,c^{2} d x \,{\mathrm e}^{2 i \left (x b +a \right )}-3 i c^{2} d \,{\mathrm e}^{2 i \left (x b +a \right )}+3 i d^{3} x^{2}+2 b \,c^{3} {\mathrm e}^{2 i \left (x b +a \right )}+6 i c \,d^{2} x +3 i c^{2} d}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {3 d^{2} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{3}}-\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {3 d^{2} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}-\frac {3 i d^{3} x^{2}}{b^{2}}-\frac {6 i d^{3} x a}{b^{3}}-\frac {3 i d^{3} a^{2}}{b^{4}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{3}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{4}}-\frac {3 i d^{3} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{3}}-\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 d^{3} a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {3 d^{3} a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{4}}\) | \(409\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (98) = 196\).
Time = 0.27 (sec) , antiderivative size = 591, normalized size of antiderivative = 5.14 \[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=\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 ) - 3 \, {\left (i \, d^{3} \cos \left (b x + a\right )^{2} - i \, d^{3}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, d^{3} \cos \left (b x + a\right )^{2} + i \, d^{3}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, d^{3} \cos \left (b x + a\right )^{2} + i \, d^{3}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (i \, d^{3} \cos \left (b x + a\right )^{2} - 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 )}} \]
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\[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=\int \left (c + d x\right )^{3} \cos {\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (98) = 196\).
Time = 0.49 (sec) , antiderivative size = 1044, normalized size of antiderivative = 9.08 \[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Too large to display} \]
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\[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{{\sin \left (a+b\,x\right )}^3} \,d x \]
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