Optimal. Leaf size=1051 \[ -\frac {b e f x}{2 a^2 d}-\frac {\left (a^2-b^2\right ) e f x}{2 a^2 b d}-\frac {b f^2 x^2}{4 a^2 d}-\frac {\left (a^2-b^2\right ) f^2 x^2}{4 a^2 b d}+\frac {i b (e+f x)^3}{3 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {2 \left (a^2-b^2\right ) f (e+f x) \cos (c+d x)}{a b^2 d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a^2 d^2}+\frac {2 \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 )}{a^2 b^3 d^3}+\frac {2 \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 )}{a^2 b^3 d^3}-\frac {b f^2 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {2 \left (a^2-b^2\right ) f^2 \sin (c+d x)}{a b^2 d^3}-\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \sin (c+d x)}{a b^2 d}+\frac {b f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a^2 d^2}+\frac {\left (a^2-b^2\right ) f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a^2 b d^2}-\frac {b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac {\left (a^2-b^2\right ) f^2 \sin ^2(c+d x)}{4 a^2 b d^3}+\frac {b (e+f x)^2 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \sin ^2(c+d x)}{2 a^2 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.40, antiderivative size = 1051, normalized size of antiderivative = 1.00, number of steps
used = 60, number of rules used = 20, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4639, 4493,
3392, 3377, 2717, 2713, 4495, 4268, 2317, 2438, 4490, 3391, 4489, 3798, 2221, 2611, 2320, 6724,
4621, 4615} \begin {gather*} -\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}+\frac {i b (e+f x)^3}{3 a^2 f}+\frac {b \sin ^2(c+d x) (e+f x)^2}{2 a^2 d}+\frac {\left (a^2-b^2\right ) \sin ^2(c+d x) (e+f x)^2}{2 a^2 b d}-\frac {\csc (c+d x) (e+f x)^2}{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)^2}{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)^2}{a^2 b^3 d}-\frac {b \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^2}{a^2 d}-\frac {\left (a^2-b^2\right ) \sin (c+d x) (e+f x)^2}{a b^2 d}-\frac {\sin (c+d x) (e+f x)^2}{a d}-\frac {4 f \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)}{a d^2}-\frac {2 \left (a^2-b^2\right ) f \cos (c+d x) (e+f x)}{a b^2 d^2}-\frac {2 f \cos (c+d x) (e+f x)}{a d^2}-\frac {2 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)}{a^2 b^3 d^2}-\frac {2 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)}{a^2 b^3 d^2}+\frac {i b f \text {PolyLog}\left (2,e^{2 i (c+d x)}\right ) (e+f x)}{a^2 d^2}+\frac {b f \cos (c+d x) \sin (c+d x) (e+f x)}{2 a^2 d^2}+\frac {\left (a^2-b^2\right ) f \cos (c+d x) \sin (c+d x) (e+f x)}{2 a^2 b d^2}-\frac {b f^2 x^2}{4 a^2 d}-\frac {\left (a^2-b^2\right ) f^2 x^2}{4 a^2 b d}-\frac {b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac {\left (a^2-b^2\right ) f^2 \sin ^2(c+d x)}{4 a^2 b d^3}-\frac {b e f x}{2 a^2 d}-\frac {\left (a^2-b^2\right ) e f x}{2 a^2 b d}+\frac {2 i f^2 \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {2 \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 )}{a^2 b^3 d^3}+\frac {2 \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 )}{a^2 b^3 d^3}-\frac {b f^2 \text {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {2 \left (a^2-b^2\right ) f^2 \sin (c+d x)}{a b^2 d^3}+\frac {2 f^2 \sin (c+d x)}{a d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 2713
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3798
Rule 4268
Rule 4489
Rule 4490
Rule 4493
Rule 4495
Rule 4615
Rule 4621
Rule 4639
Rule 6724
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {\int (e+f x)^2 \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \cos ^4(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 a d^2}-\frac {(e+f x)^2 \cos ^2(c+d x) \sin (c+d x)}{3 a d}-\frac {2 \int (e+f x)^2 \cos (c+d x) \, dx}{3 a}-\frac {\int (e+f x)^2 \cos (c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \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)^2 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {\left (2 f^2\right ) \int \cos ^3(c+d x) \, dx}{9 a d^2}\\ &=-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {5 (e+f x)^2 \sin (c+d x)}{3 a d}+\frac {2 \int (e+f x)^2 \cos (c+d x) \, dx}{3 a}-\frac {b \int (e+f x)^2 \cot (c+d x) \, dx}{a^2}+\frac {b \int (e+f x)^2 \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)^2 \cos (c+d x) \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x)^2 \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)^2 \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {(4 f) \int (e+f x) \sin (c+d x) \, dx}{3 a d}+\frac {(2 f) \int (e+f x) \csc (c+d x) \, dx}{a d}+\frac {(2 f) \int (e+f x) \sin (c+d x) \, dx}{a d}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{9 a d^3}-\frac {\left (2 f^2\right ) \int \cos ^3(c+d x) \, dx}{9 a d^2}\\ &=\frac {i b (e+f x)^3}{3 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {10 f (e+f x) \cos (c+d x)}{3 a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {2 f^2 \sin (c+d x)}{9 a d^3}-\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin (c+d x)}{b^2 d}+\frac {b (e+f x)^2 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin ^2(c+d x)}{2 b d}-\frac {2 f^2 \sin ^3(c+d x)}{27 a d^3}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{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)^2}{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)^2}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{b^2}-\frac {(4 f) \int (e+f x) \sin (c+d x) \, dx}{3 a d}-\frac {(b f) \int (e+f x) \sin ^2(c+d x) \, dx}{a^2 d}+\frac {\left (2 a \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x) \sin (c+d x) \, dx}{b^2 d}-\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x) \sin ^2(c+d x) \, dx}{b d}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{9 a d^3}+\frac {\left (4 f^2\right ) \int \cos (c+d x) \, dx}{3 a d^2}+\frac {\left (2 f^2\right ) \int \cos (c+d x) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d^2}\\ &=\frac {i b (e+f x)^3}{3 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {10 f^2 \sin (c+d x)}{3 a d^3}-\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin (c+d x)}{b^2 d}+\frac {b f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x) \sin (c+d x)}{2 b d^2}-\frac {b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^2 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin ^2(c+d x)}{2 b d}-\frac {(b f) \int (e+f x) \, dx}{2 a^2 d}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}-\frac {\left (2 \left (a^2-b^2\right )^2 f\right ) \int (e+f x) \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 (2 \left (a^2-b^2\right )^2 f\right ) \int (e+f x) \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 (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x) \, dx}{2 b d}+\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}-\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}-\frac {\left (4 f^2\right ) \int \cos (c+d x) \, dx}{3 a d^2}+\frac {\left (2 a \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int \cos (c+d x) \, dx}{b^2 d^2}\\ &=-\frac {b e f x}{2 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) e f x}{2 b d}-\frac {b f^2 x^2}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 x^2}{4 b d}+\frac {i b (e+f x)^3}{3 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a^2 d^2}+\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f^2 \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin (c+d x)}{b^2 d}+\frac {b f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x) \sin (c+d x)}{2 b d^2}-\frac {b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^2 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin ^2(c+d x)}{2 b d}-\frac {\left (i b f^2\right ) \int \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a^2 d^2}+\frac {\left (2 i \left (a^2-b^2\right )^2 f^2\right ) \int \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 (2 i \left (a^2-b^2\right )^2 f^2\right ) \int \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 {b e f x}{2 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) e f x}{2 b d}-\frac {b f^2 x^2}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 x^2}{4 b d}+\frac {i b (e+f x)^3}{3 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a^2 d^2}+\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f^2 \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin (c+d x)}{b^2 d}+\frac {b f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x) \sin (c+d x)}{2 b d^2}-\frac {b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^2 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin ^2(c+d x)}{2 b d}-\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {\left (2 \left (a^2-b^2\right )^2 f^2\right ) \text {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^2 b^3 d^3}+\frac {\left (2 \left (a^2-b^2\right )^2 f^2\right ) \text {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^2 b^3 d^3}\\ &=-\frac {b e f x}{2 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) e f x}{2 b d}-\frac {b f^2 x^2}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 x^2}{4 b d}+\frac {i b (e+f x)^3}{3 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^3}{3 a^2 b^3 f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}-\frac {2 i \left (a^2-b^2\right )^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^2 b^3 d^2}+\frac {i b f (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a^2 d^2}+\frac {2 \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 )}{a^2 b^3 d^3}+\frac {2 \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 )}{a^2 b^3 d^3}-\frac {b f^2 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {2 a \left (1-\frac {b^2}{a^2}\right ) f^2 \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin (c+d x)}{b^2 d}+\frac {b f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f (e+f x) \cos (c+d x) \sin (c+d x)}{2 b d^2}-\frac {b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac {\left (1-\frac {b^2}{a^2}\right ) f^2 \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^2 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^2 \sin ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(5228\) vs. \(2(1051)=2102\).
time = 10.38, size = 5228, normalized size = 4.97 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.36, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\cot ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 3146 vs. \(2 (978) = 1956\).
time = 0.77, size = 3146, normalized size = 2.99 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\left (e + f x\right )^{2} \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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