Integrand size = 18, antiderivative size = 137 \[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=-b^2 c x-\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {i a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \cot (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac {b^2 d \log (\sin (e+f x))}{f^2}-\frac {i a b d \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2} \]
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Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3803, 3798, 2221, 2317, 2438, 3801, 3556} \[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=\frac {a^2 (c+d x)^2}{2 d}+\frac {2 a b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i a b (c+d x)^2}{d}-\frac {i a b d \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x) \cot (e+f x)}{f}-b^2 c x+\frac {b^2 d \log (\sin (e+f x))}{f^2}-\frac {1}{2} b^2 d x^2 \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3556
Rule 3798
Rule 3801
Rule 3803
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (c+d x)+2 a b (c+d x) \cot (e+f x)+b^2 (c+d x) \cot ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \cot (e+f x) \, dx+b^2 \int (c+d x) \cot ^2(e+f x) \, dx \\ & = \frac {a^2 (c+d x)^2}{2 d}-\frac {i a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \cot (e+f x)}{f}-(4 i a b) \int \frac {e^{2 i (e+f x)} (c+d x)}{1-e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x) \, dx+\frac {\left (b^2 d\right ) \int \cot (e+f x) \, dx}{f} \\ & = -b^2 c x-\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {i a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \cot (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac {b^2 d \log (\sin (e+f x))}{f^2}-\frac {(2 a b d) \int \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f} \\ & = -b^2 c x-\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {i a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \cot (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac {b^2 d \log (\sin (e+f x))}{f^2}+\frac {(i a b d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^2} \\ & = -b^2 c x-\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {i a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \cot (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac {b^2 d \log (\sin (e+f x))}{f^2}-\frac {i a b d \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2} \\ \end{align*}
Time = 8.52 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.74 \[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=\frac {(a+b \cot (e+f x))^2 \sin (e+f x) \left (-2 b^2 f (c+d x) \cos (e+f x)-(a-b) (a+b) (e+f x) (-2 c f+d (e-f x)) \sin (e+f x)+2 b^2 d (\log (\cos (e+f x))+\log (\tan (e+f x))) \sin (e+f x)-4 a b d e (\log (\cos (e+f x))+\log (\tan (e+f x))) \sin (e+f x)+4 a b c f (\log (\cos (e+f x))+\log (\tan (e+f x))) \sin (e+f x)+4 a b d \left ((e+f x) \log \left (1-e^{2 i (e+f x)}\right )-\frac {1}{2} i \left ((e+f x)^2+\operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )\right )\right ) \sin (e+f x)\right )}{2 f^2 (b \cos (e+f x)+a \sin (e+f x))^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (127 ) = 254\).
Time = 0.72 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.66
method | result | size |
risch | \(-\frac {b^{2} d \,x^{2}}{2}-\frac {2 i b^{2} \left (d x +c \right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}+\frac {a^{2} d \,x^{2}}{2}-b^{2} c x -\frac {2 i b a d \operatorname {polylog}\left (2, {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+a^{2} c x -\frac {4 i b a d e x}{f}+\frac {b^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {2 b^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {b^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f^{2}}+\frac {2 b a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}-\frac {4 b a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {2 b a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}+\frac {4 b e a d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 b e a d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f^{2}}-i a b d \,x^{2}+2 i a b c x -\frac {2 i b a d \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {2 b a d \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}+\frac {2 b a d \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}-\frac {2 i b a d \,e^{2}}{f^{2}}+\frac {2 b a d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{f}\) | \(365\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (124) = 248\).
Time = 0.28 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.76 \[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=-\frac {2 \, b^{2} d f x + i \, a b d {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) - i \, a b d {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) + 2 \, b^{2} c f + {\left (2 \, a b d e - 2 \, a b c f - b^{2} d\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) \sin \left (2 \, f x + 2 \, e\right ) + {\left (2 \, a b d e - 2 \, a b c f - b^{2} d\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) - \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, {\left (a b d f x + a b d e\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, {\left (a b d f x + a b d e\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) \sin \left (2 \, f x + 2 \, e\right ) + 2 \, {\left (b^{2} d f x + b^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) - {\left ({\left (a^{2} - b^{2}\right )} d f^{2} x^{2} + 2 \, {\left (a^{2} - b^{2}\right )} c f^{2} x\right )} \sin \left (2 \, f x + 2 \, e\right )}{2 \, f^{2} \sin \left (2 \, f x + 2 \, e\right )} \]
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\[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=\int \left (a + b \cot {\left (e + f x \right )}\right )^{2} \left (c + d x\right )\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (124) = 248\).
Time = 0.34 (sec) , antiderivative size = 774, normalized size of antiderivative = 5.65 \[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=\frac {2 \, {\left (f x + e\right )} a^{2} c + \frac {{\left (f x + e\right )}^{2} a^{2} d}{f} - \frac {2 \, {\left (f x + e\right )} a^{2} d e}{f} + 4 \, a b c \log \left (\sin \left (f x + e\right )\right ) - \frac {4 \, a b d e \log \left (\sin \left (f x + e\right )\right )}{f} + \frac {2 \, {\left ({\left (2 \, a b - i \, b^{2}\right )} {\left (f x + e\right )}^{2} d + 4 \, b^{2} d e - 4 \, b^{2} c f - 2 \, {\left (-i \, b^{2} d e + i \, b^{2} c f\right )} {\left (f x + e\right )} - 2 \, {\left (2 \, {\left (f x + e\right )} a b d + b^{2} d - {\left (2 \, {\left (f x + e\right )} a b d + b^{2} d\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (-2 i \, {\left (f x + e\right )} a b d - i \, b^{2} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) + 2 \, {\left (b^{2} d \cos \left (2 \, f x + 2 \, e\right ) + i \, b^{2} d \sin \left (2 \, f x + 2 \, e\right ) - b^{2} d\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) - 1\right ) - 4 \, {\left ({\left (f x + e\right )} a b d \cos \left (2 \, f x + 2 \, e\right ) + i \, {\left (f x + e\right )} a b d \sin \left (2 \, f x + 2 \, e\right ) - {\left (f x + e\right )} a b d\right )} \arctan \left (\sin \left (f x + e\right ), -\cos \left (f x + e\right ) + 1\right ) - {\left ({\left (2 \, a b - i \, b^{2}\right )} {\left (f x + e\right )}^{2} d + 2 \, {\left (i \, b^{2} d e - i \, b^{2} c f + 2 \, b^{2} d\right )} {\left (f x + e\right )}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (a b d \cos \left (2 \, f x + 2 \, e\right ) + i \, a b d \sin \left (2 \, f x + 2 \, e\right ) - a b d\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) - 4 \, {\left (a b d \cos \left (2 \, f x + 2 \, e\right ) + i \, a b d \sin \left (2 \, f x + 2 \, e\right ) - a b d\right )} {\rm Li}_2\left (e^{\left (i \, f x + i \, e\right )}\right ) + {\left (2 i \, {\left (f x + e\right )} a b d + i \, b^{2} d + {\left (-2 i \, {\left (f x + e\right )} a b d - i \, b^{2} d\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (2 \, {\left (f x + e\right )} a b d + b^{2} d\right )} \sin \left (2 \, f x + 2 \, 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 ) + {\left (2 i \, {\left (f x + e\right )} a b d + i \, b^{2} d + {\left (-2 i \, {\left (f x + e\right )} a b d - i \, b^{2} d\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (2 \, {\left (f x + e\right )} a b d + b^{2} d\right )} \sin \left (2 \, f x + 2 \, 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 ) + {\left ({\left (-2 i \, a b - b^{2}\right )} {\left (f x + e\right )}^{2} d + 2 \, {\left (b^{2} d e - b^{2} c f - 2 i \, b^{2} d\right )} {\left (f x + e\right )}\right )} \sin \left (2 \, f x + 2 \, e\right )\right )}}{-2 i \, f \cos \left (2 \, f x + 2 \, e\right ) + 2 \, f \sin \left (2 \, f x + 2 \, e\right ) + 2 i \, f}}{2 \, f} \]
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\[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=\int { {\left (d x + c\right )} {\left (b \cot \left (f x + e\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (c+d x) (a+b \cot (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right ) \,d x \]
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