Integrand size = 34, antiderivative size = 517 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {e x}{a}+\frac {\left (1-\frac {a^2}{b^2}\right ) e x}{a}-\frac {f x^2}{2 a}+\frac {\left (1-\frac {a^2}{b^2}\right ) f x^2}{2 a}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a^2 b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (a^2-b^2\right ) f \sin (c+d x)}{a^2 b d^2} \]
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Time = 1.38 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {4639, 4493, 3391, 3801, 3556, 4490, 2713, 3377, 2717, 4268, 2317, 2438, 4621, 3404, 2296, 2221} \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {f \left (a^2-b^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {f \left (a^2-b^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {f \left (a^2-b^2\right ) \sin (c+d x)}{a^2 b d^2}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b^2 d}-\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a^2 b d}+\frac {e x \left (1-\frac {a^2}{b^2}\right )}{a}+\frac {f x^2 \left (1-\frac {a^2}{b^2}\right )}{2 a}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {e x}{a}-\frac {f x^2}{2 a} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2713
Rule 2717
Rule 3377
Rule 3391
Rule 3404
Rule 3556
Rule 3801
Rule 4268
Rule 4490
Rule 4493
Rule 4621
Rule 4639
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \cos ^2(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = -\frac {\int (e+f x) \cos ^2(c+d x) \, dx}{a}+\frac {\int (e+f x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos ^3(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {f \cos ^2(c+d x)}{4 a d^2}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int (e+f x) \, dx}{2 a}-\frac {\int (e+f x) \, dx}{a}+\frac {\int (e+f x) \cos ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos (c+d x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {f \int \cot (c+d x) \, dx}{a d} \\ & = -\frac {3 e x}{2 a}-\frac {3 f x^2}{4 a}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {f \log (\sin (c+d x))}{a d^2}+\frac {\int (e+f x) \, dx}{2 a}-\frac {b \int (e+f x) \csc (c+d x) \, dx}{a^2}+\frac {b \int (e+f x) \sin (c+d x) \, dx}{a^2}-\frac {\left (a \left (1-\frac {b^2}{a^2}\right )\right ) \int (e+f x) \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x) \sin (c+d x) \, dx}{b}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {e+f x}{a+b \sin (c+d x)} \, dx}{b^2} \\ & = -\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {f \log (\sin (c+d x))}{a d^2}+\frac {\left (2 \left (a^2-b^2\right )^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2 b^2}+\frac {(b f) \int \cos (c+d x) \, dx}{a^2 d}+\frac {(b f) \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \cos (c+d x) \, dx}{b d} \\ & = -\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {f \log (\sin (c+d x))}{a d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}-\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}+\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {(i b f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^2} \\ & = -\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}+\frac {\left (i \left (a^2-b^2\right )^{3/2} f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d}-\frac {\left (i \left (a^2-b^2\right )^{3/2} f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d} \\ & = -\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}+\frac {\left (\left (a^2-b^2\right )^{3/2} f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a-2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^2}-\frac {\left (\left (a^2-b^2\right )^{3/2} f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a+2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^2} \\ & = -\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1091\) vs. \(2(517)=1034\).
Time = 12.25 (sec) , antiderivative size = 1091, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a (c+d x) (2 d e-2 c f+f (c+d x))}{2 b^2 d^2}-\frac {(d e-c f+f (c+d x)) \cos (c+d x)}{b d^2}+\frac {\left (-d e \cos \left (\frac {1}{2} (c+d x)\right )+c f \cos \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{2 a d^2}-\frac {b e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d}+\frac {b c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d^2}+\frac {f (\log (\cos (c+d x))+\log (\tan (c+d x)))}{a d^2}-\frac {b f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )\right )\right )}{a^2 d^2}+\frac {\left (a^2-b^2\right )^2 (d e+d f x) \left (\frac {2 (d e-c f) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (-\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}\right )}{a^2 b^2 d^2 \left (d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (d e \sin \left (\frac {1}{2} (c+d x)\right )-c f \sin \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}+\frac {f \sin (c+d x)}{b d^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. 5309 vs. \(2 (475 ) = 950\).
Time = 2.82 (sec) , antiderivative size = 5310, normalized size of antiderivative = 10.27
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1751 vs. \(2 (462) = 924\).
Time = 0.55 (sec) , antiderivative size = 1751, normalized size of antiderivative = 3.39 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]
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