Integrand size = 36, antiderivative size = 1144 \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \cos (c+d x)}{a^2 b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x)^3 \cos (c+d x)}{a^2 b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \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)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a^2 b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f (e+f x)^2 \sin (c+d x)}{a^2 b d^2} \]
[Out]
Time = 1.78 (sec) , antiderivative size = 1144, normalized size of antiderivative = 1.00, number of steps used = 66, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4639, 4493, 3392, 32, 3391, 3801, 3798, 2221, 2611, 2320, 6724, 4490, 3377, 2717, 2713, 4268, 6744, 4621, 3404, 2296} \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {(e+f x)^4}{4 a f}+\frac {2 b \text {arctanh}\left (e^{i (c+d x)}\right ) (e+f x)^3}{a^2 d}-\frac {b \cos (c+d x) (e+f x)^3}{a^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^3}{a^2 b d}-\frac {\cot (c+d x) (e+f x)^3}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b^2 d}-\frac {i (e+f x)^3}{a d}+\frac {3 f \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)^2}{a^2 d^2}+\frac {3 i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)^2}{a^2 d^2}-\frac {3 \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 ) (e+f x)^2}{a^2 b^2 d^2}+\frac {3 \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 ) (e+f x)^2}{a^2 b^2 d^2}+\frac {3 b f \sin (c+d x) (e+f x)^2}{a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)^2}{a^2 b d^2}+\frac {6 b f^2 \cos (c+d x) (e+f x)}{a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 \cos (c+d x) (e+f x)}{a^2 b d^3}-\frac {3 i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) (e+f x)}{a^2 d^3}-\frac {6 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) (e+f x)}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a^2 b d^4} \]
[In]
[Out]
Rule 32
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 2713
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3404
Rule 3798
Rule 3801
Rule 4268
Rule 4490
Rule 4493
Rule 4621
Rule 4639
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = -\frac {\int (e+f x)^3 \cos ^2(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos ^3(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int (e+f x)^3 \, dx}{2 a}-\frac {\int (e+f x)^3 \, dx}{a}+\frac {\int (e+f x)^3 \cos ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos (c+d x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {(3 f) \int (e+f x)^2 \cot (c+d x) \, dx}{a d}+\frac {\left (3 f^2\right ) \int (e+f x) \cos ^2(c+d x) \, dx}{2 a d^2} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {3 (e+f x)^4}{8 a f}+\frac {3 f^3 \cos ^2(c+d x)}{8 a d^4}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}+\frac {\int (e+f x)^3 \, dx}{2 a}-\frac {b \int (e+f x)^3 \csc (c+d x) \, dx}{a^2}+\frac {b \int (e+f x)^3 \sin (c+d x) \, dx}{a^2}-\frac {\left (a \left (1-\frac {b^2}{a^2}\right )\right ) \int (e+f x)^3 \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x)^3 \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)^3}{a+b \sin (c+d x)} \, dx}{b^2}-\frac {(6 i f) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1-e^{2 i (c+d x)}} \, dx}{a d}+\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2}-\frac {\left (3 f^2\right ) \int (e+f x) \cos ^2(c+d x) \, dx}{2 a d^2} \\ & = \frac {3 e f^2 x}{4 a d^2}+\frac {3 f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {\left (2 \left (a^2-b^2\right )^2\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2 b^2}+\frac {(3 b f) \int (e+f x)^2 \cos (c+d x) \, dx}{a^2 d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \cos (c+d x) \, dx}{b d}-\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \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)^3}{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)^3}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}+\frac {\left (6 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{a^2 d^2}-\frac {\left (6 \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{b d^2}+\frac {\left (3 i f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \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)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (3 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x)^2 \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 (3 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x)^2 \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 (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {\left (6 b f^3\right ) \int \cos (c+d x) \, dx}{a^2 d^3}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx}{a^2 d^3}-\frac {\left (6 \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int \cos (c+d x) \, dx}{b d^3} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \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)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^2}-\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^2}+\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^4}-\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^4} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \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)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^3}-\frac {\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d^3} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \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)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^4}-\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^4} \\ & = -\frac {i (e+f x)^3}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^4}{4 b^2 f}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {6 b f^2 (e+f x) \cos (c+d x)}{a^2 d^3}+\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {b (e+f x)^3 \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \cos (c+d x)}{b d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \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)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 \left (a^2-b^2\right )^{3/2} f (e+f x)^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 {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^4}-\frac {6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac {6 \left (1-\frac {b^2}{a^2}\right ) f^3 \sin (c+d x)}{b d^4}+\frac {3 b f (e+f x)^2 \sin (c+d x)}{a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \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. \(3915\) vs. \(2(1144)=2288\).
Time = 8.23 (sec) , antiderivative size = 3915, normalized size of antiderivative = 3.42 \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \left (\cot ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]
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Exception generated. \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]
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