Integrand size = 26, antiderivative size = 937 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 i a (e+f x)^3 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\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 \operatorname {PolyLog}\left (4,\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 \operatorname {PolyLog}\left (4,-e^{2 i (c+d x)}\right )}{4 \left (a^2-b^2\right ) d^4} \]
[Out]
Time = 1.06 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {4629, 4615, 2221, 2611, 6744, 2320, 6724, 6874, 4266, 3800} \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {6 i a \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac {6 i a \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}-\frac {6 i b \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \arctan \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} \]
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4266
Rule 4615
Rule 4629
Rule 6724
Rule 6744
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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 \arctan \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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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) \operatorname {PolyLog}\left (2,\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) \operatorname {PolyLog}\left (2,\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 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,\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 \operatorname {PolyLog}\left (3,\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 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}-\frac {\left (6 i b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\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 \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}-\frac {\left (6 a f^3\right ) \int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3} \\ & = -\frac {2 i a (e+f x)^3 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\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 \operatorname {PolyLog}\left (4,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \operatorname {PolyLog}\left (3,-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 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\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 \operatorname {PolyLog}\left (4,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \arctan \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 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2}-\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {6 a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,\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) \operatorname {PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,\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 \operatorname {PolyLog}\left (4,\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 \operatorname {PolyLog}\left (4,-e^{2 i (c+d x)}\right )}{4 \left (a^2-b^2\right ) d^4} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2496\) vs. \(2(937)=1874\).
Time = 6.88 (sec) , antiderivative size = 2496, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{3} \sec \left (d x +c \right )}{a +b \sin \left (d x +c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3069 vs. \(2 (822) = 1644\).
Time = 0.62 (sec) , antiderivative size = 3069, normalized size of antiderivative = 3.28 \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \sec {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sec \left (d x + c\right )}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]
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