Integrand size = 34, antiderivative size = 566 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^3}{3 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}-\frac {2 f (e+f x) \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {2 f^2 \sin (c+d x)}{b d^3}-\frac {(e+f x)^2 \sin (c+d x)}{b d} \]
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Time = 1.24 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {4639, 4493, 4489, 3391, 3798, 2221, 2611, 2320, 6724, 4621, 3377, 2717, 4615} \[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 f^2 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {2 f^2 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^3}-\frac {2 i f \left (a^2-b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {2 i f \left (a^2-b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a b^2 d}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac {(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {i (e+f x)^3}{3 a f}+\frac {2 f^2 \sin (c+d x)}{b d^3}-\frac {2 f (e+f x) \cos (c+d x)}{b d^2}-\frac {(e+f x)^2 \sin (c+d x)}{b d} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3391
Rule 3798
Rule 4489
Rule 4493
Rule 4615
Rule 4621
Rule 4639
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \cos ^2(c+d x) \cot (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = \frac {\int (e+f x)^2 \cot (c+d x) \, dx}{a}-\frac {\int (e+f x)^2 \cos (c+d x) \, dx}{b}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {(e+f x)^2 \cos (c+d x)}{a+b \sin (c+d x)} \, dx \\ & = -\frac {i (e+f x)^3}{3 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}-\frac {(e+f x)^2 \sin (c+d x)}{b d}-\frac {(2 i) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1-e^{2 i (c+d x)}} \, dx}{a}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {e^{i (c+d x)} (e+f x)^2}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {e^{i (c+d x)} (e+f x)^2}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx+\frac {(2 f) \int (e+f x) \sin (c+d x) \, dx}{b d} \\ & = -\frac {i (e+f x)^3}{3 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}-\frac {2 f (e+f x) \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {(e+f x)^2 \sin (c+d x)}{b d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d}+\frac {\left (2 \left (-\frac {a}{b}+\frac {b}{a}\right ) f\right ) \int (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d}+\frac {\left (2 \left (-\frac {a}{b}+\frac {b}{a}\right ) f\right ) \int (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d}+\frac {\left (2 f^2\right ) \int \cos (c+d x) \, dx}{b d^2} \\ & = -\frac {i (e+f x)^3}{3 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}-\frac {2 f (e+f x) \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac {2 f^2 \sin (c+d x)}{b d^3}-\frac {(e+f x)^2 \sin (c+d x)}{b d}+\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 i \left (a^2-b^2\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a b^2 d^2}+\frac {\left (2 i \left (a^2-b^2\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a b^2 d^2} \\ & = -\frac {i (e+f x)^3}{3 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}-\frac {2 f (e+f x) \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac {2 f^2 \sin (c+d x)}{b d^3}-\frac {(e+f x)^2 \sin (c+d x)}{b d}+\frac {f^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {\left (2 \left (a^2-b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a b^2 d^3}+\frac {\left (2 \left (a^2-b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a b^2 d^3} \\ & = -\frac {i (e+f x)^3}{3 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a b^2 f}-\frac {2 f (e+f x) \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^2}+\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {2 f^2 \sin (c+d x)}{b d^3}-\frac {(e+f x)^2 \sin (c+d x)}{b d} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1741\) vs. \(2(566)=1132\).
Time = 8.60 (sec) , antiderivative size = 1741, normalized size of antiderivative = 3.08 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {e^{i c} f^2 \csc (c) \left (2 d^3 e^{-2 i c} x^3+3 i d^2 \left (1-e^{-2 i c}\right ) x^2 \log \left (1-e^{-i (c+d x)}\right )+3 i d^2 \left (1-e^{-2 i c}\right ) x^2 \log \left (1+e^{-i (c+d x)}\right )-6 d \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-6 d \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+6 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )+6 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )\right )}{6 a d^3}+\frac {\left (a^2-b^2\right ) \left (-6 i d^3 e^2 e^{2 i c} x-6 i d^3 e e^{2 i c} f x^2-2 i d^3 e^{2 i c} f^2 x^3-3 d^2 e^2 \log \left (b-2 i a e^{i (c+d x)}-b e^{2 i (c+d x)}\right )+3 d^2 e^2 e^{2 i c} \log \left (b-2 i a e^{i (c+d x)}-b e^{2 i (c+d x)}\right )-6 d^2 e f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 d^2 e e^{2 i c} f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-3 d^2 f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+3 d^2 e^{2 i c} f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 d^2 e f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 d^2 e e^{2 i c} f x \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-3 d^2 f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+3 d^2 e^{2 i c} f^2 x^2 \log \left (1+\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 i d \left (-1+e^{2 i c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (2 c+d x)}}{a e^{i c}+i \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 i d \left (-1+e^{2 i c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (2 c+d x)}}{a e^{i c}+i \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 e^{2 i c} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (2 c+d x)}}{a e^{i c}+i \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )-6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+6 e^{2 i c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (2 c+d x)}}{i a e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )\right )}{3 a b^2 d^3 \left (-1+e^{2 i c}\right )}+\frac {a x \left (3 e^2+3 e f x+f^2 x^2\right ) \cos (c) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{6 b^2}-\frac {\cos (d x) \left (2 d e f \cos (c)+2 d f^2 x \cos (c)+d^2 e^2 \sin (c)-2 f^2 \sin (c)+2 d^2 e f x \sin (c)+d^2 f^2 x^2 \sin (c)\right )}{b d^3}+\frac {e^2 \csc (c) (-d x \cos (c)+\log (\cos (d x) \sin (c)+\cos (c) \sin (d x)) \sin (c))}{a d \left (\cos ^2(c)+\sin ^2(c)\right )}-\frac {\left (d^2 e^2 \cos (c)-2 f^2 \cos (c)+2 d^2 e f x \cos (c)+d^2 f^2 x^2 \cos (c)-2 d e f \sin (c)-2 d f^2 x \sin (c)\right ) \sin (d x)}{b d^3}-\frac {e f \csc (c) \sec (c) \left (d^2 e^{i \arctan (\tan (c))} x^2+\frac {\left (i d x (-\pi +2 \arctan (\tan (c)))-\pi \log \left (1+e^{-2 i d x}\right )-2 (d x+\arctan (\tan (c))) \log \left (1-e^{2 i (d x+\arctan (\tan (c)))}\right )+\pi \log (\cos (d x))+2 \arctan (\tan (c)) \log (\sin (d x+\arctan (\tan (c))))+i \operatorname {PolyLog}\left (2,e^{2 i (d x+\arctan (\tan (c)))}\right )\right ) \tan (c)}{\sqrt {1+\tan ^2(c)}}\right )}{a d^2 \sqrt {\sec ^2(c) \left (\cos ^2(c)+\sin ^2(c)\right )}} \]
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\[\int \frac {\left (f x +e \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{a +b \sin \left (d x +c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2260 vs. \(2 (511) = 1022\).
Time = 0.53 (sec) , antiderivative size = 2260, normalized size of antiderivative = 3.99 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cos ^{2}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]
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