Optimal. Leaf size=1138 \[ \frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {2 \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^3}{a d}+\frac {\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^3}{a b^2 d}+\frac {\cos (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 b^3 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 b^3 d}-\frac {\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}-\frac {3 f \cos ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac {3 i f \text {Li}_2\left (-e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 i f \text {Li}_2\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)^2}{a b^2 d^2}-\frac {3 f \sin (c+d x) (e+f x)^2}{a d^2}-\frac {6 \left (a^2-b^2\right ) f^2 \cos (c+d x) (e+f x)}{a b^2 d^3}-\frac {6 f^2 \cos (c+d x) (e+f x)}{a d^3}-\frac {6 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 f^2 \text {Li}_3\left (e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}+\frac {3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}+\frac {3 f^3 x^2}{8 b d^2}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}+\frac {3 e f^2 x}{4 b d^2}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^4}+\frac {6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a b^2 d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4} \]
[Out]
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Rubi [A] time = 2.11, antiderivative size = 1138, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 18, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {4543, 4408, 4405, 3311, 3296, 2637, 2633, 4183, 2531, 6609, 2282, 6589, 4525, 32, 3310, 3323, 2264, 2190} \[ \frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {(e+f x)^4}{8 b f}-\frac {2 \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^3}{a d}+\frac {\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^3}{a b^2 d}+\frac {\cos (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 b^3 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 b^3 d}-\frac {\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}-\frac {3 f \cos ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac {3 i f \text {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 i f \text {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)^2}{a b^2 d^2}-\frac {3 f \sin (c+d x) (e+f x)^2}{a d^2}-\frac {6 \left (a^2-b^2\right ) f^2 \cos (c+d x) (e+f x)}{a b^2 d^3}-\frac {6 f^2 \cos (c+d x) (e+f x)}{a d^3}-\frac {6 f^2 \text {PolyLog}\left (3,-e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 f^2 \text {PolyLog}\left (3,e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}+\frac {3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}+\frac {3 f^3 x^2}{8 b d^2}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}+\frac {3 e f^2 x}{4 b d^2}-\frac {6 i f^3 \text {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^4}+\frac {6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a b^2 d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2633
Rule 2637
Rule 3296
Rule 3310
Rule 3311
Rule 3323
Rule 4183
Rule 4405
Rule 4408
Rule 4525
Rule 4543
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cos ^3(c+d x) \cot (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^3 \cos (c+d x) \cot (c+d x) \, dx}{a}-\frac {\int (e+f x)^3 \cos ^2(c+d x) \, dx}{b}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\\ &=-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\frac {\int (e+f x)^3 \sin (c+d x) \, dx}{a}+\left (\frac {1}{a}-\frac {a}{b^2}\right ) \int (e+f x)^3 \sin (c+d x) \, dx-\frac {\int (e+f x)^3 \, dx}{2 b}+\frac {\left (a^2-b^2\right ) \int (e+f x)^3 \, dx}{b^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {(e+f x)^3}{a+b \sin (c+d x)} \, dx}{a b^3}+\frac {\left (3 f^2\right ) \int (e+f x) \cos ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\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 b^3}-\frac {(3 f) \int (e+f x)^2 \cos (c+d x) \, dx}{a d}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}+\frac {\left (3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f\right ) \int (e+f x)^2 \cos (c+d x) \, dx}{d}+\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}+\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 b^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 b^2}-\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{a d^2}-\frac {\left (6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{d^2}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\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 b^3 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 b^3 d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b 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 b^3 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 b^3 d}+\frac {\left (6 f^3\right ) \int \cos (c+d x) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^3\right ) \int \cos (c+d x) \, dx}{d^3}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\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 b^3 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 b^3 d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a b^3 d^2}+\frac {\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a b^3 d^2}-\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\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 b^3 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 b^3 d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a b^3 d^3}+\frac {\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a b^3 d^3}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\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 b^3 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 b^3 d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (6 \left (a^2-b^2\right )^{3/2} 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 b^3 d^4}+\frac {\left (6 \left (a^2-b^2\right )^{3/2} 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 b^3 d^4}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}-\frac {(e+f x)^4}{8 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac {3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\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 b^3 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 b^3 d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {3 \left (a^2-b^2\right )^{3/2} 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 b^3 d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i \left (a^2-b^2\right )^{3/2} 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 b^3 d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^4}+\frac {6 \left (a^2-b^2\right )^{3/2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {6 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 6.73, size = 1181, normalized size = 1.04 \[ \frac {2 a \left (2 a^2-3 b^2\right ) f^3 x^4 d^4+8 a \left (2 a^2-3 b^2\right ) e f^2 x^3 d^4+12 a \left (2 a^2-3 b^2\right ) e^2 f x^2 d^4+8 a \left (2 a^2-3 b^2\right ) e^3 x d^4-32 b^3 (e+f x)^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^3+16 a^2 b (e+f x)^3 \cos (c+d x) d^3-4 a b^2 (e+f x)^3 \sin (2 (c+d x)) d^3-6 a b^2 f (e+f x)^2 \cos (2 (c+d x)) d^2+48 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text {Li}_2\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right ) d^2-48 a^2 b f (e+f x)^2 \sin (c+d x) d^2-96 a^2 b f^2 (e+f x) \cos (c+d x) d+6 a b^2 f^2 (e+f x) \sin (2 (c+d x)) d+3 a b^2 f^3 \cos (2 (c+d x))+16 i \left (a^2-b^2\right )^{3/2} \left (2 i e^3 \tan ^{-1}\left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right ) d^3+f^3 x^3 \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right ) d^3+3 e^2 f x \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right ) d^3-f^3 x^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^3-3 e f^2 x^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^3-3 e^2 f x \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^3+3 i f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^2+6 f^2 (e+f x) \text {Li}_3\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right ) d-6 e f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d-6 f^3 x \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d+6 i f^3 \text {Li}_4\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right )-6 i f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )\right )+48 i b^3 f \left (-2 \text {Li}_4(-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text {Li}_3(-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text {Li}_2(-\cos (c+d x)-i \sin (c+d x))\right )-48 i b^3 f \left (-2 \text {Li}_4(\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text {Li}_3(\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text {Li}_2(\cos (c+d x)+i \sin (c+d x))\right )+96 a^2 b f^3 \sin (c+d x)}{16 a b^3 d^4} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 1.18, size = 4221, normalized size = 3.71 \[ \text {result too large to display} \]
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 = 6.87, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{a +b \sin \left (d x +c \right )}\, dx \]
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 \[ \text {Exception raised: ValueError} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{3} \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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