Optimal. Leaf size=147 \[ \frac {a (c+d x)^4}{4 d}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b (c+d x)^4}{4 d}+\frac {3 i b d^3 \text {Li}_4\left (e^{2 i (e+f x)}\right )}{4 f^4} \]
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Rubi [A] time = 0.26, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3722, 3717, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 b d^2 (c+d x) \text {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 i b d (c+d x)^2 \text {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 i b d^3 \text {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}+\frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b (c+d x)^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 3722
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \cot (e+f x)) \, dx &=\int \left (a (c+d x)^3+b (c+d x)^3 \cot (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^4}{4 d}+b \int (c+d x)^3 \cot (e+f x) \, dx\\ &=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}-(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{1-e^{2 i (e+f x)}} \, dx\\ &=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {(3 b d) \int (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac {\left (3 i b d^2\right ) \int (c+d x) \text {Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac {\left (3 b d^3\right ) \int \text {Li}_3\left (e^{2 i (e+f x)}\right ) \, dx}{2 f^3}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}+\frac {\left (3 i b d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 f^4}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b d (c+d x)^2 \text {Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 i b d^3 \text {Li}_4\left (e^{2 i (e+f x)}\right )}{4 f^4}\\ \end {align*}
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Mathematica [B] time = 7.50, size = 730, normalized size = 4.97 \[ \frac {1}{4} x \csc (e) \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) (a \sin (e)+b \cos (e))+\frac {b c^3 \csc (e) (\sin (e) \log (\sin (e) \cos (f x)+\cos (e) \sin (f x))-f x \cos (e))}{f \left (\sin ^2(e)+\cos ^2(e)\right )}-\frac {3 b c^2 d \csc (e) \sec (e) \left (f^2 x^2 e^{i \tan ^{-1}(\tan (e))}+\frac {\tan (e) \left (i \text {Li}_2\left (e^{2 i \left (f x+\tan ^{-1}(\tan (e))\right )}\right )+i f x \left (2 \tan ^{-1}(\tan (e))-\pi \right )-2 \left (\tan ^{-1}(\tan (e))+f x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (e))+f x\right )}\right )+2 \tan ^{-1}(\tan (e)) \log \left (\sin \left (\tan ^{-1}(\tan (e))+f x\right )\right )-\pi \log \left (1+e^{-2 i f x}\right )+\pi \log (\cos (f x))\right )}{\sqrt {\tan ^2(e)+1}}\right )}{2 f^2 \sqrt {\sec ^2(e) \left (\sin ^2(e)+\cos ^2(e)\right )}}-\frac {b c d^2 e^{i e} \csc (e) \left (2 e^{-2 i e} f^3 x^3+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )-6 e^{-2 i e} \left (-1+e^{2 i e}\right ) \left (f x \text {Li}_2\left (-e^{-i (e+f x)}\right )-i \text {Li}_3\left (-e^{-i (e+f x)}\right )\right )-6 e^{-2 i e} \left (-1+e^{2 i e}\right ) \left (f x \text {Li}_2\left (e^{-i (e+f x)}\right )-i \text {Li}_3\left (e^{-i (e+f x)}\right )\right )\right )}{2 f^3}-\frac {b d^3 e^{i e} \csc (e) \left (e^{-2 i e} f^4 x^4+2 i \left (1-e^{-2 i e}\right ) f^3 x^3 \log \left (1-e^{-i (e+f x)}\right )+2 i \left (1-e^{-2 i e}\right ) f^3 x^3 \log \left (1+e^{-i (e+f x)}\right )-6 e^{-2 i e} \left (-1+e^{2 i e}\right ) \left (f^2 x^2 \text {Li}_2\left (-e^{-i (e+f x)}\right )-2 i f x \text {Li}_3\left (-e^{-i (e+f x)}\right )-2 \text {Li}_4\left (-e^{-i (e+f x)}\right )\right )-6 e^{-2 i e} \left (-1+e^{2 i e}\right ) \left (f^2 x^2 \text {Li}_2\left (e^{-i (e+f x)}\right )-2 i f x \text {Li}_3\left (e^{-i (e+f x)}\right )-2 \text {Li}_4\left (e^{-i (e+f x)}\right )\right )\right )}{4 f^4} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.74, size = 626, normalized size = 4.26 \[ \frac {2 \, a d^{3} f^{4} x^{4} + 8 \, a c d^{2} f^{4} x^{3} + 12 \, a c^{2} d f^{4} x^{2} + 8 \, a c^{3} f^{4} x + 3 i \, b d^{3} {\rm polylog}\left (4, \cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) - 3 i \, b d^{3} {\rm polylog}\left (4, \cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + {\left (-6 i \, b d^{3} f^{2} x^{2} - 12 i \, b c d^{2} f^{2} x - 6 i \, b c^{2} d f^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + {\left (6 i \, b d^{3} f^{2} x^{2} + 12 i \, b c d^{2} f^{2} x + 6 i \, b c^{2} d f^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) - 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) - 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) - \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) + 6 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + 6 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right )}{8 \, f^{4}} \]
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 \[ \int {\left (d x + c\right )}^{3} {\left (b \cot \left (f x + e\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.32, size = 857, normalized size = 5.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.16, size = 966, normalized size = 6.57 \[ \text {result too large to display} \]
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 \[ \int \left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3 \,d x \]
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 \[ \int \left (a + b \cot {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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