Optimal. Leaf size=1432 \[ -\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}+\frac {i b (e+f x)^4}{4 a^2 f}+\frac {b \sin ^2(c+d x) (e+f x)^3}{2 a^2 d}+\frac {\left (a^2-b^2\right ) \sin ^2(c+d x) (e+f x)^3}{2 a^2 b d}-\frac {\csc (c+d x) (e+f x)^3}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^3}{a^2 d}-\frac {\left (a^2-b^2\right ) \sin (c+d x) (e+f x)^3}{a b^2 d}-\frac {\sin (c+d x) (e+f x)^3}{a d}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x)^3}{4 a^2 b d}-\frac {6 f \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 \left (a^2-b^2\right ) f \cos (c+d x) (e+f x)^2}{a b^2 d^2}-\frac {3 f \cos (c+d x) (e+f x)^2}{a d^2}-\frac {3 i \left (a^2-b^2\right )^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b^3 d^2}+\frac {3 i b f \text {Li}_2\left (e^{2 i (c+d x)}\right ) (e+f x)^2}{2 a^2 d^2}+\frac {3 b f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 b d^2}-\frac {3 b f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 d^3}-\frac {3 \left (a^2-b^2\right ) f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 b d^3}+\frac {6 i f^2 \text {Li}_2\left (-e^{i (c+d x)}\right ) (e+f x)}{a d^3}-\frac {6 i f^2 \text {Li}_2\left (e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^3 d^3}-\frac {3 b f^2 \text {Li}_3\left (e^{2 i (c+d x)}\right ) (e+f x)}{2 a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 \sin (c+d x) (e+f x)}{a b^2 d^3}+\frac {6 f^2 \sin (c+d x) (e+f x)}{a d^3}+\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (a^2-b^2\right ) f^3 x}{8 a^2 b d^3}+\frac {6 \left (a^2-b^2\right ) f^3 \cos (c+d x)}{a b^2 d^4}+\frac {6 f^3 \cos (c+d x)}{a d^4}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}-\frac {3 i b f^3 \text {Li}_4\left (e^{2 i (c+d x)}\right )}{4 a^2 d^4}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (a^2-b^2\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 b d^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.95, antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps used = 85, number of rules used = 21, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4543, 4408, 3311, 3296, 2638, 3310, 4410, 4183, 2531, 2282, 6589, 4405, 32, 2635, 8, 4404, 3717, 2190, 6609, 4525, 4519} \[ -\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}+\frac {i b (e+f x)^4}{4 a^2 f}+\frac {b \sin ^2(c+d x) (e+f x)^3}{2 a^2 d}+\frac {\left (a^2-b^2\right ) \sin ^2(c+d x) (e+f x)^3}{2 a^2 b d}-\frac {\csc (c+d x) (e+f x)^3}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^3}{a^2 d}-\frac {\left (a^2-b^2\right ) \sin (c+d x) (e+f x)^3}{a b^2 d}-\frac {\sin (c+d x) (e+f x)^3}{a d}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x)^3}{4 a^2 b d}-\frac {6 f \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 \left (a^2-b^2\right ) f \cos (c+d x) (e+f x)^2}{a b^2 d^2}-\frac {3 f \cos (c+d x) (e+f x)^2}{a d^2}-\frac {3 i \left (a^2-b^2\right )^2 f \text {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^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f \text {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^3 d^2}+\frac {3 i b f \text {PolyLog}\left (2,e^{2 i (c+d x)}\right ) (e+f x)^2}{2 a^2 d^2}+\frac {3 b f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 b d^2}-\frac {3 b f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 d^3}-\frac {3 \left (a^2-b^2\right ) f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 b d^3}+\frac {6 i f^2 \text {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)}{a d^3}-\frac {6 i f^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^3 d^3}-\frac {3 b f^2 \text {PolyLog}\left (3,e^{2 i (c+d x)}\right ) (e+f x)}{2 a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 \sin (c+d x) (e+f x)}{a b^2 d^3}+\frac {6 f^2 \sin (c+d x) (e+f x)}{a d^3}+\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (a^2-b^2\right ) f^3 x}{8 a^2 b d^3}+\frac {6 \left (a^2-b^2\right ) f^3 \cos (c+d x)}{a b^2 d^4}+\frac {6 f^3 \cos (c+d x)}{a d^4}-\frac {6 f^3 \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}-\frac {3 i b f^3 \text {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a^2 d^4}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (a^2-b^2\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 b d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 32
Rule 2190
Rule 2282
Rule 2531
Rule 2635
Rule 2638
Rule 3296
Rule 3310
Rule 3311
Rule 3717
Rule 4183
Rule 4404
Rule 4405
Rule 4408
Rule 4410
Rule 4519
Rule 4525
Rule 4543
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {\int (e+f x)^3 \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos ^4(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac {f (e+f x)^2 \cos ^3(c+d x)}{3 a d^2}-\frac {(e+f x)^3 \cos ^2(c+d x) \sin (c+d x)}{3 a d}-\frac {2 \int (e+f x)^3 \cos (c+d x) \, dx}{3 a}-\frac {\int (e+f x)^3 \cos (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos ^2(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 ^3(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {\left (2 f^2\right ) \int (e+f x) \cos ^3(c+d x) \, dx}{3 a d^2}\\ &=\frac {2 f^3 \cos ^3(c+d x)}{27 a d^4}-\frac {(e+f x)^3 \csc (c+d x)}{a d}-\frac {5 (e+f x)^3 \sin (c+d x)}{3 a d}+\frac {2 f^2 (e+f x) \cos ^2(c+d x) \sin (c+d x)}{9 a d^3}+\frac {2 \int (e+f x)^3 \cos (c+d x) \, dx}{3 a}-\frac {b \int (e+f x)^3 \cot (c+d x) \, dx}{a^2}+\frac {b \int (e+f x)^3 \cos (c+d x) \sin (c+d x) \, dx}{a^2}-\frac {\left (a \left (1-\frac {b^2}{a^2}\right )\right ) \int (e+f x)^3 \cos (c+d x) \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x)^3 \cos (c+d 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)^3 \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {(2 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \csc (c+d x) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}+\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{9 a d^2}-\frac {\left (2 f^2\right ) \int (e+f x) \cos ^3(c+d x) \, dx}{3 a d^2}\\ &=\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {5 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {4 f^2 (e+f x) \sin (c+d x)}{9 a d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)^3}{1-e^{2 i (c+d x)}} \, dx}{a^2}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{b^2}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{b^2}-\frac {(2 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}-\frac {(3 b f) \int (e+f x)^2 \sin ^2(c+d x) \, dx}{2 a^2 d}+\frac {\left (3 a \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \sin (c+d x) \, dx}{b^2 d}-\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \sin ^2(c+d x) \, dx}{2 b d}-\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{9 a d^2}+\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{9 a d^3}\\ &=\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {4 f^3 \cos (c+d x)}{9 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}+\frac {10 f^2 (e+f x) \sin (c+d x)}{a d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac {(3 b f) \int (e+f x)^2 \, dx}{4 a^2 d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}-\frac {\left (3 \left (a^2-b^2\right )^2 f\right ) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d}-\frac {\left (3 \left (a^2-b^2\right )^2 f\right ) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d}-\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \, dx}{4 b d}-\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}+\frac {\left (6 a \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{b^2 d^2}-\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{9 a d^3}-\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}+\frac {\left (3 b f^3\right ) \int \sin ^2(c+d x) \, dx}{4 a^2 d^3}+\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int \sin ^2(c+d x) \, dx}{4 b d^3}\\ &=-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {10 f^3 \cos (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac {\left (3 i b f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a^2 d^2}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^2}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^2}-\frac {\left (6 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (6 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}+\frac {\left (3 b f^3\right ) \int 1 \, dx}{8 a^2 d^3}-\frac {\left (6 a \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int \sin (c+d x) \, dx}{b^2 d^3}+\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int 1 \, dx}{8 b d^3}\\ &=\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 x}{8 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x)}{b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}+\frac {\left (3 b f^3\right ) \int \text {Li}_3\left (e^{2 i (c+d x)}\right ) \, dx}{2 a^2 d^3}-\frac {\left (6 \left (a^2-b^2\right )^2 f^3\right ) \int \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^3}-\frac {\left (6 \left (a^2-b^2\right )^2 f^3\right ) \int \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^3}\\ &=\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 x}{8 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x)}{b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac {\left (3 i b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{4 a^2 d^4}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^3 d^4}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^3 d^4}\\ &=\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 x}{8 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x)}{b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^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^3 d}+\frac {\left (a^2-b^2\right )^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^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}-\frac {3 i b f^3 \text {Li}_4\left (e^{2 i (c+d x)}\right )}{4 a^2 d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [B] time = 49.23, size = 3944, normalized size = 2.75 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.37, size = 4936, normalized size = 3.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 11.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\cot ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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