Optimal. Leaf size=112 \[ \frac {a (c+d x)^3}{3 d}-\frac {i b d (c+d x) \text {Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b (c+d x)^3}{3 d}+\frac {b d^2 \text {Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3} \]
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Rubi [A] time = 0.21, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3722, 3717, 2190, 2531, 2282, 6589} \[ -\frac {i b d (c+d x) \text {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {b d^2 \text {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {a (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b (c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 3722
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \cot (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \cot (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \cot (e+f x) \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {i b (c+d x)^3}{3 d}-(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {i b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {(2 b d) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {i b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b d (c+d x) \text {Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac {\left (i b d^2\right ) \int \text {Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {i b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b d (c+d x) \text {Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {i b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b d (c+d x) \text {Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac {b d^2 \text {Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}\\ \end {align*}
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Mathematica [B] time = 2.57, size = 406, normalized size = 3.62 \[ \frac {3 a c^2 f^3 x+3 a c d f^3 x^2+a d^2 f^3 x^3+3 b c^2 f^2 \log (\sin (e+f x))+3 b c d f^3 x^2 \cot (e)-3 b c d f^3 x^2 e^{i \tan ^{-1}(\tan (e))} \cot (e) \sqrt {\sec ^2(e)}-6 i b c d f^2 x \tan ^{-1}(\tan (e))+6 b c d f^2 x \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (e))+f x\right )}\right )-3 i b c d f \text {Li}_2\left (e^{2 i \left (f x+\tan ^{-1}(\tan (e))\right )}\right )+6 b c d f \tan ^{-1}(\tan (e)) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (e))+f x\right )}\right )-6 b c d f \tan ^{-1}(\tan (e)) \log \left (\sin \left (\tan ^{-1}(\tan (e))+f x\right )\right )+3 i \pi b c d f^2 x+3 \pi b c d f \log \left (1+e^{-2 i f x}\right )-3 \pi b c d f \log (\cos (f x))+3 b d^2 f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 b d^2 f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )+6 i b d^2 f x \text {Li}_2\left (-e^{-i (e+f x)}\right )+6 i b d^2 f x \text {Li}_2\left (e^{-i (e+f x)}\right )+6 b d^2 \text {Li}_3\left (-e^{-i (e+f x)}\right )+6 b d^2 \text {Li}_3\left (e^{-i (e+f x)}\right )+i b d^2 f^3 x^3}{3 f^3} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.61, size = 403, normalized size = 3.60 \[ \frac {4 \, a d^{2} f^{3} x^{3} + 12 \, a c d f^{3} x^{2} + 12 \, a c^{2} f^{3} x + 3 \, b d^{2} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + 3 \, b d^{2} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + {\left (-6 i \, b d^{2} f x - 6 i \, b c d f\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + {\left (6 i \, b d^{2} f x + 6 i \, b c d f\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + 6 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) + 6 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) - \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) + 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) + 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right )}{12 \, f^{3}} \]
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 \[ \int {\left (d x + c\right )}^{2} {\left (b \cot \left (f x + e\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.22, size = 516, normalized size = 4.61 \[ a \,c^{2} x +a c d \,x^{2}+\frac {2 b c d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{f}+\frac {2 b c d \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}+\frac {4 b c d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i b c d \polylog \left (2, {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i b c d \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i b \,d^{2} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {2 i b \,d^{2} \polylog \left (2, {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}+\frac {2 b \,d^{2} \polylog \left (3, -{\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {2 b \,d^{2} \polylog \left (3, {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {b \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f^{3}}+i b \,c^{2} x -\frac {i b \,d^{2} x^{3}}{3}+\frac {b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}-\frac {2 b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}-\frac {b \,d^{2} \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e^{2}}{f^{3}}+\frac {b \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x^{2}}{f}+\frac {a \,d^{2} x^{3}}{3}-\frac {4 i b c d e x}{f}+\frac {4 i b \,d^{2} e^{3}}{3 f^{3}}-\frac {2 b \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-i b c d \,x^{2}+\frac {b \,d^{2} \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{f}+\frac {2 i b \,d^{2} e^{2} x}{f^{2}}-\frac {2 b c d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f^{2}}+\frac {2 b c d \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}-\frac {2 i b c d \,e^{2}}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.75, size = 520, normalized size = 4.64 \[ \frac {6 \, {\left (f x + e\right )} a c^{2} + \frac {2 \, {\left (f x + e\right )}^{3} a d^{2}}{f^{2}} - \frac {6 \, {\left (f x + e\right )}^{2} a d^{2} e}{f^{2}} + \frac {6 \, {\left (f x + e\right )} a d^{2} e^{2}}{f^{2}} + \frac {6 \, {\left (f x + e\right )}^{2} a c d}{f} - \frac {12 \, {\left (f x + e\right )} a c d e}{f} + 6 \, b c^{2} \log \left (\sin \left (f x + e\right )\right ) + \frac {6 \, b d^{2} e^{2} \log \left (\sin \left (f x + e\right )\right )}{f^{2}} - \frac {12 \, b c d e \log \left (\sin \left (f x + e\right )\right )}{f} + \frac {-2 i \, {\left (f x + e\right )}^{3} b d^{2} + 12 \, b d^{2} {\rm Li}_{3}(-e^{\left (i \, f x + i \, e\right )}) + 12 \, b d^{2} {\rm Li}_{3}(e^{\left (i \, f x + i \, e\right )}) + {\left (6 i \, b d^{2} e - 6 i \, b c d f\right )} {\left (f x + e\right )}^{2} + {\left (6 i \, {\left (f x + e\right )}^{2} b d^{2} + {\left (-12 i \, b d^{2} e + 12 i \, b c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) + {\left (-6 i \, {\left (f x + e\right )}^{2} b d^{2} + {\left (12 i \, b d^{2} e - 12 i \, b c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (\sin \left (f x + e\right ), -\cos \left (f x + e\right ) + 1\right ) + {\left (-12 i \, {\left (f x + e\right )} b d^{2} + 12 i \, b d^{2} e - 12 i \, b c d f\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) + {\left (-12 i \, {\left (f x + e\right )} b d^{2} + 12 i \, b d^{2} e - 12 i \, b c d f\right )} {\rm Li}_2\left (e^{\left (i \, f x + i \, e\right )}\right ) + 3 \, {\left ({\left (f x + e\right )}^{2} b d^{2} - 2 \, {\left (b d^{2} e - b c d f\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + 3 \, {\left ({\left (f x + e\right )}^{2} b d^{2} - 2 \, {\left (b d^{2} e - b c d f\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )}{f^{2}}}{6 \, f} \]
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 \[ \int \left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2 \,d x \]
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 \[ \int \left (a + b \cot {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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