Optimal. Leaf size=1051 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.24098, antiderivative size = 1051, normalized size of antiderivative = 1., 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 = {4543, 4408, 3311, 3296, 2637, 2633, 4410, 4183, 2279, 2391, 4405, 3310, 4404, 3717, 2190, 2531, 2282, 6589, 4525, 4519} \[ -\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} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4543
Rule 4408
Rule 3311
Rule 3296
Rule 2637
Rule 2633
Rule 4410
Rule 4183
Rule 2279
Rule 2391
Rule 4405
Rule 3310
Rule 4404
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 4525
Rule 4519
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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \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^2 b^3 d^3}+\frac{\left (2 \left (a^2-b^2\right )^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^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*}
Mathematica [B] time = 13.5755, size = 5156, normalized size = 4.91 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 4.864, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 5.62446, size = 7509, normalized size = 7.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]