Optimal. Leaf size=825 \[ \frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d}+\frac {\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^2}{a b^2 d}+\frac {\cos (c+d x) (e+f x)^2}{a d}+\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a b^3 d}-\frac {\cos (c+d x) \sin (c+d x) (e+f x)^2}{2 b d}-\frac {f \cos ^2(c+d x) (e+f x)}{2 b d^2}+\frac {2 i f \text {Li}_2\left (-e^{i (c+d x)}\right ) (e+f x)}{a d^2}-\frac {2 i f \text {Li}_2\left (e^{i (c+d x)}\right ) (e+f x)}{a d^2}+\frac {2 \left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a b^3 d^2}-\frac {2 \left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a b^3 d^2}-\frac {2 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)}{a b^2 d^2}-\frac {2 f \sin (c+d x) (e+f x)}{a d^2}+\frac {f^2 x}{4 b d^2}-\frac {2 \left (a^2-b^2\right ) f^2 \cos (c+d x)}{a b^2 d^3}-\frac {2 f^2 \cos (c+d x)}{a d^3}-\frac {2 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (a^2-b^2\right )^{3/2} f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^3}-\frac {2 i \left (a^2-b^2\right )^{3/2} f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^3}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3} \]
[Out]
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Rubi [A] time = 1.63, antiderivative size = 825, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 18, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {4543, 4408, 4405, 3310, 3296, 2638, 4183, 2531, 2282, 6589, 4525, 3311, 32, 2635, 8, 3323, 2264, 2190} \[ \frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d}+\frac {\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^2}{a b^2 d}+\frac {\cos (c+d x) (e+f x)^2}{a d}+\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a b^3 d}-\frac {\cos (c+d x) \sin (c+d x) (e+f x)^2}{2 b d}-\frac {f \cos ^2(c+d x) (e+f x)}{2 b d^2}+\frac {2 i f \text {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)}{a d^2}-\frac {2 i f \text {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)}{a d^2}+\frac {2 \left (a^2-b^2\right )^{3/2} f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a b^3 d^2}-\frac {2 \left (a^2-b^2\right )^{3/2} f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a b^3 d^2}-\frac {2 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)}{a b^2 d^2}-\frac {2 f \sin (c+d x) (e+f x)}{a d^2}+\frac {f^2 x}{4 b d^2}-\frac {2 \left (a^2-b^2\right ) f^2 \cos (c+d x)}{a b^2 d^3}-\frac {2 f^2 \cos (c+d x)}{a d^3}-\frac {2 f^2 \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (a^2-b^2\right )^{3/2} f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^3}-\frac {2 i \left (a^2-b^2\right )^{3/2} f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^3}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2635
Rule 2638
Rule 3296
Rule 3310
Rule 3311
Rule 3323
Rule 4183
Rule 4405
Rule 4408
Rule 4525
Rule 4543
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \cos ^3(c+d x) \cot (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^2 \cos (c+d x) \cot (c+d x) \, dx}{a}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \, dx}{b}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\\ &=-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int (e+f x)^2 \csc (c+d x) \, dx}{a}-\frac {\int (e+f x)^2 \sin (c+d x) \, dx}{a}+\left (\frac {1}{a}-\frac {a}{b^2}\right ) \int (e+f x)^2 \sin (c+d x) \, dx-\frac {\int (e+f x)^2 \, dx}{2 b}+\frac {\left (a^2-b^2\right ) \int (e+f x)^2 \, dx}{b^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {(e+f x)^2}{a+b \sin (c+d x)} \, dx}{a b^3}+\frac {f^2 \int \cos ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^2 \cos (c+d x)}{d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \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)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a b^3}-\frac {(2 f) \int (e+f x) \cos (c+d x) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}+\frac {\left (2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f\right ) \int (e+f x) \cos (c+d x) \, dx}{d}+\frac {f^2 \int 1 \, dx}{4 b d^2}\\ &=\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^2 \cos (c+d x)}{d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x) \sin (c+d x)}{d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \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)^2}{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)^2}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a b^2}-\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \sin (c+d x) \, dx}{a d^2}-\frac {\left (2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2\right ) \int \sin (c+d x) \, dx}{d^2}\\ &=\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 \cos (c+d x)}{d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^2 \cos (c+d x)}{d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^2 \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)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x) \sin (c+d x)}{d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (2 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x) \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 (2 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x) \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 (2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}+\frac {\left (2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}\\ &=\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 \cos (c+d x)}{d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^2 \cos (c+d x)}{d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^2 \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)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \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 {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \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 {2 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x) \sin (c+d x)}{d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (2 \left (a^2-b^2\right )^{3/2} f^2\right ) \int \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 (2 \left (a^2-b^2\right )^{3/2} f^2\right ) \int \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 {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 \cos (c+d x)}{d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^2 \cos (c+d x)}{d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^2 \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)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \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 {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \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 {2 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x) \sin (c+d x)}{d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\left (2 i \left (a^2-b^2\right )^{3/2} f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\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^3}-\frac {\left (2 i \left (a^2-b^2\right )^{3/2} f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\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^3}\\ &=\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f^2 \cos (c+d x)}{d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x)^2 \cos (c+d x)}{d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^2 \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)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \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 {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \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 {2 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {2 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 )}{a b^3 d^3}-\frac {2 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 )}{a b^3 d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {2 \left (\frac {1}{a}-\frac {a}{b^2}\right ) f (e+f x) \sin (c+d x)}{d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 5.08, size = 1254, normalized size = 1.52 \[ -\frac {-24 d^2 e^2 \log \left (1-e^{i (c+d x)}\right ) b^3-24 d^2 f^2 x^2 \log \left (1-e^{i (c+d x)}\right ) b^3-48 d^2 e f x \log \left (1-e^{i (c+d x)}\right ) b^3+24 d^2 e^2 \log \left (1+e^{i (c+d x)}\right ) b^3+24 d^2 f^2 x^2 \log \left (1+e^{i (c+d x)}\right ) b^3+48 d^2 e f x \log \left (1+e^{i (c+d x)}\right ) b^3-48 i d e f \text {Li}_2\left (-e^{i (c+d x)}\right ) b^3-48 i d f^2 x \text {Li}_2\left (-e^{i (c+d x)}\right ) b^3+48 i d e f \text {Li}_2\left (e^{i (c+d x)}\right ) b^3+48 i d f^2 x \text {Li}_2\left (e^{i (c+d x)}\right ) b^3+48 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right ) b^3-48 f^2 \text {Li}_3\left (e^{i (c+d x)}\right ) b^3+12 a d^3 f^2 x^3 b^2+36 a d^3 e f x^2 b^2+36 a d^3 e^2 x b^2+6 a d e f \cos (2 (c+d x)) b^2+6 a d f^2 x \cos (2 (c+d x)) b^2+6 a d^2 e^2 \sin (2 (c+d x)) b^2-3 a f^2 \sin (2 (c+d x)) b^2+6 a d^2 f^2 x^2 \sin (2 (c+d x)) b^2+12 a d^2 e f x \sin (2 (c+d x)) b^2-24 a^2 d^2 e^2 \cos (c+d x) b+48 a^2 f^2 \cos (c+d x) b-24 a^2 d^2 f^2 x^2 \cos (c+d x) b-48 a^2 d^2 e f x \cos (c+d x) b+48 a^2 d e f \sin (c+d x) b+48 a^2 d f^2 x \sin (c+d x) b-8 a^3 d^3 f^2 x^3-24 a^3 d^3 e f x^2-24 a^3 d^3 e^2 x+48 \left (a^2-b^2\right )^{3/2} d^2 e^2 \tan ^{-1}\left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )-24 i \left (a^2-b^2\right )^{3/2} d^2 f^2 x^2 \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right )-48 i \left (a^2-b^2\right )^{3/2} d^2 e f x \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right )+24 i \left (a^2-b^2\right )^{3/2} d^2 f^2 x^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )+48 i \left (a^2-b^2\right )^{3/2} d^2 e f x \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )-48 \left (a^2-b^2\right )^{3/2} d e f \text {Li}_2\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right )-48 \left (a^2-b^2\right )^{3/2} d f^2 x \text {Li}_2\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right )+48 \left (a^2-b^2\right )^{3/2} d e f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )+48 \left (a^2-b^2\right )^{3/2} d f^2 x \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )-48 i \left (a^2-b^2\right )^{3/2} f^2 \text {Li}_3\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right )+48 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 )}{24 a b^3 d^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.87, size = 2797, normalized size = 3.39 \[ \text {result too large to display} \]
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 = 6.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \left (\cos ^{3}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{a +b \sin \left (d x +c \right )}\, dx \]
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 \[ \int \frac {\left (e + f x\right )^{2} \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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