Optimal. Leaf size=937 \[ -\frac{6 i a \text{PolyLog}\left (4,-i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{6 i a \text{PolyLog}\left (4,i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac{6 i b \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac{6 i b \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{3 i b \text{PolyLog}\left (4,-e^{2 i (c+d x)}\right ) f^3}{4 \left (a^2-b^2\right ) d^4}-\frac{6 a (e+f x) \text{PolyLog}\left (3,-i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac{6 a (e+f x) \text{PolyLog}\left (3,i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{6 b (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{6 b (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac{3 b (e+f x) \text{PolyLog}\left (3,-e^{2 i (c+d x)}\right ) f^2}{2 \left (a^2-b^2\right ) d^3}+\frac{3 i a (e+f x)^2 \text{PolyLog}\left (2,-i e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}-\frac{3 i a (e+f x)^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right ) d^2}-\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right ) f}{2 \left (a^2-b^2\right ) d^2}-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.61759, antiderivative size = 937, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {4533, 4519, 2190, 2531, 6609, 2282, 6589, 6742, 4181, 3719} \[ -\frac{6 i a \text{PolyLog}\left (4,-i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{6 i a \text{PolyLog}\left (4,i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac{6 i b \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac{6 i b \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{3 i b \text{PolyLog}\left (4,-e^{2 i (c+d x)}\right ) f^3}{4 \left (a^2-b^2\right ) d^4}-\frac{6 a (e+f x) \text{PolyLog}\left (3,-i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac{6 a (e+f x) \text{PolyLog}\left (3,i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{6 b (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{6 b (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac{3 b (e+f x) \text{PolyLog}\left (3,-e^{2 i (c+d x)}\right ) f^2}{2 \left (a^2-b^2\right ) d^3}+\frac{3 i a (e+f x)^2 \text{PolyLog}\left (2,-i e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}-\frac{3 i a (e+f x)^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right ) d^2}-\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right ) f}{2 \left (a^2-b^2\right ) d^2}-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4533
Rule 4519
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 6742
Rule 4181
Rule 3719
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \sec (c+d x) (a-b \sin (c+d x)) \, dx}{a^2-b^2}-\frac{b^2 \int \frac{(e+f x)^3 \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{a^2-b^2}\\ &=\frac{i b (e+f x)^4}{4 \left (a^2-b^2\right ) f}+\frac{\int \left (a (e+f x)^3 \sec (c+d x)-b (e+f x)^3 \tan (c+d x)\right ) \, dx}{a^2-b^2}-\frac{b^2 \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}{a^2-b^2}-\frac{b^2 \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}{a^2-b^2}\\ &=\frac{i b (e+f x)^4}{4 \left (a^2-b^2\right ) f}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{a \int (e+f x)^3 \sec (c+d x) \, dx}{a^2-b^2}-\frac{b \int (e+f x)^3 \tan (c+d x) \, dx}{a^2-b^2}+\frac{(3 b f) \int (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d}+\frac{(3 b f) \int (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d}\\ &=-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\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-b^2}-\frac{(3 a f) \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}+\frac{(3 a f) \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}-\frac{\left (6 i b 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}{\left (a^2-b^2\right ) d^2}-\frac{\left (6 i b 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}{\left (a^2-b^2\right ) d^2}\\ &=-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac{3 i a f (e+f x)^2 \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i a f (e+f x)^2 \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac{(3 b f) \int (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}-\frac{\left (6 i a f^2\right ) \int (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac{\left (6 i a f^2\right ) \int (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac{\left (6 b f^3\right ) \int \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d^3}+\frac{\left (6 b f^3\right ) \int \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d^3}\\ &=-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac{3 i a f (e+f x)^2 \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i a f (e+f x)^2 \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac{6 a f^2 (e+f x) \text{Li}_3\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{6 a f^2 (e+f x) \text{Li}_3\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac{\left (3 i b f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}-\frac{\left (6 i b 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 )}{\left (a^2-b^2\right ) d^4}-\frac{\left (6 i b 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 )}{\left (a^2-b^2\right ) d^4}+\frac{\left (6 a f^3\right ) \int \text{Li}_3\left (-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}-\frac{\left (6 a f^3\right ) \int \text{Li}_3\left (i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}\\ &=-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac{3 i a f (e+f x)^2 \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i a f (e+f x)^2 \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac{6 a f^2 (e+f x) \text{Li}_3\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{6 a f^2 (e+f x) \text{Li}_3\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b f^2 (e+f x) \text{Li}_3\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac{6 i b f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac{\left (6 i a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{\left (6 i a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac{\left (3 b f^3\right ) \int \text{Li}_3\left (-e^{2 i (c+d x)}\right ) \, dx}{2 \left (a^2-b^2\right ) d^3}\\ &=-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac{3 i a f (e+f x)^2 \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i a f (e+f x)^2 \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac{6 a f^2 (e+f x) \text{Li}_3\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{6 a f^2 (e+f x) \text{Li}_3\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b f^2 (e+f x) \text{Li}_3\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac{6 i a f^3 \text{Li}_4\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{6 i a f^3 \text{Li}_4\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}+\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 \left (a^2-b^2\right ) d^4}\\ &=-\frac{2 i a (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac{b (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac{b (e+f x)^3 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac{3 i a f (e+f x)^2 \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i a f (e+f x)^2 \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac{6 a f^2 (e+f x) \text{Li}_3\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{6 a f^2 (e+f x) \text{Li}_3\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b f^2 (e+f x) \text{Li}_3\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac{6 i a f^3 \text{Li}_4\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{6 i a f^3 \text{Li}_4\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^4}+\frac{3 i b f^3 \text{Li}_4\left (-e^{2 i (c+d x)}\right )}{4 \left (a^2-b^2\right ) d^4}\\ \end{align*}
Mathematica [B] time = 9.83598, size = 2496, normalized size = 2.66 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.914, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\sec \left ( dx+c \right ) }{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 3.97027, size = 7549, normalized size = 8.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right )^{3} \sec{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \sec \left (d x + c\right )}{b \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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