Integrand size = 28, antiderivative size = 802 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2} \]
[Out]
Time = 0.96 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {4611, 3392, 32, 3391, 3377, 2717, 3404, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {(e+f x)^4}{8 b f}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {a \cos (c+d x) (e+f x)^3}{b^2 d}+\frac {i a^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^3}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^3}{b^3 \sqrt {a^2-b^2} d}-\frac {\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}+\frac {3 f \sin ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac {3 a^3 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a f \sin (c+d x) (e+f x)^2}{b^2 d^2}-\frac {6 a f^2 \cos (c+d x) (e+f x)}{b^2 d^3}+\frac {6 i a^3 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{b^3 \sqrt {a^2-b^2} d^3}+\frac {3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}-\frac {3 f^3 x^2}{8 b d^2}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}-\frac {3 e f^2 x}{4 b d^2}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4} \]
[In]
[Out]
Rule 32
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3404
Rule 4611
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \sin ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac {a \int (e+f x)^3 \sin (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {\int (e+f x)^3 \, dx}{2 b}-\frac {\left (3 f^2\right ) \int (e+f x) \sin ^2(c+d x) \, dx}{2 b d^2} \\ & = \frac {(e+f x)^4}{8 b f}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}+\frac {a^2 \int (e+f x)^3 \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3}{a+b \sin (c+d x)} \, dx}{b^3}-\frac {(3 a f) \int (e+f x)^2 \cos (c+d x) \, dx}{b^2 d}-\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b^3}+\frac {\left (6 a f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{b^2 d^2} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}+\frac {\left (2 i a^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b^2 \sqrt {a^2-b^2}}-\frac {\left (2 i a^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b^2 \sqrt {a^2-b^2}}+\frac {\left (6 a f^3\right ) \int \cos (c+d x) \, dx}{b^2 d^3} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac {\left (3 i a^3 f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d}+\frac {\left (3 i a^3 f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac {\left (6 a^3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d^2}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d^2} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac {\left (6 i a^3 f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d^3}+\frac {\left (6 i a^3 f^3\right ) \int \operatorname {PolyLog}\left (3,\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d^3} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2}-\frac {\left (6 a^3 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 )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {\left (6 a^3 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 )}{b^3 \sqrt {a^2-b^2} d^4} \\ & = -\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1923\) vs. \(2(802)=1604\).
Time = 4.44 (sec) , antiderivative size = 1923, normalized size of antiderivative = 2.40 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {16 a^2 \sqrt {-\left (a^2-b^2\right )^2} d^4 e^3 x+8 b^2 \sqrt {-\left (-a^2+b^2\right )^2} d^4 e^3 x+24 a^2 \sqrt {-\left (a^2-b^2\right )^2} d^4 e^2 f x^2+12 b^2 \sqrt {-\left (-a^2+b^2\right )^2} d^4 e^2 f x^2+16 a^2 \sqrt {-\left (a^2-b^2\right )^2} d^4 e f^2 x^3+8 b^2 \sqrt {-\left (-a^2+b^2\right )^2} d^4 e f^2 x^3+4 a^2 \sqrt {-\left (a^2-b^2\right )^2} d^4 f^3 x^4+2 b^2 \sqrt {-\left (-a^2+b^2\right )^2} d^4 f^3 x^4-32 a^3 \sqrt {-a^2+b^2} d^3 e^3 \arctan \left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+16 a b \sqrt {-\left (a^2-b^2\right )^2} d^3 e^3 \cos (c+d x)-96 a b \sqrt {-\left (a^2-b^2\right )^2} d e f^2 \cos (c+d x)+48 a b \sqrt {-\left (a^2-b^2\right )^2} d^3 e^2 f x \cos (c+d x)-96 a b \sqrt {-\left (a^2-b^2\right )^2} d f^3 x \cos (c+d x)+48 a b \sqrt {-\left (a^2-b^2\right )^2} d^3 e f^2 x^2 \cos (c+d x)+16 a b \sqrt {-\left (a^2-b^2\right )^2} d^3 f^3 x^3 \cos (c+d x)-6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^2 e^2 f \cos (2 (c+d x))+3 b^2 \sqrt {-\left (a^2-b^2\right )^2} f^3 \cos (2 (c+d x))-12 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^2 e f^2 x \cos (2 (c+d x))-6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^2 f^3 x^2 \cos (2 (c+d x))-48 a^3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-48 a^3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-16 a^3 \sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+48 a^3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+48 a^3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+16 a^3 \sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+48 i a^3 \sqrt {a^2-b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-48 i a^3 \sqrt {a^2-b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-96 a^3 \sqrt {a^2-b^2} d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-96 a^3 \sqrt {a^2-b^2} d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+96 a^3 \sqrt {a^2-b^2} d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+96 a^3 \sqrt {a^2-b^2} d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-96 i a^3 \sqrt {a^2-b^2} f^3 \operatorname {PolyLog}\left (4,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+96 i a^3 \sqrt {a^2-b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-48 a b \sqrt {-\left (a^2-b^2\right )^2} d^2 e^2 f \sin (c+d x)+96 a b \sqrt {-\left (a^2-b^2\right )^2} f^3 \sin (c+d x)-96 a b \sqrt {-\left (a^2-b^2\right )^2} d^2 e f^2 x \sin (c+d x)-48 a b \sqrt {-\left (a^2-b^2\right )^2} d^2 f^3 x^2 \sin (c+d x)-4 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^3 e^3 \sin (2 (c+d x))+6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d e f^2 \sin (2 (c+d x))-12 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^3 e^2 f x \sin (2 (c+d x))+6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d f^3 x \sin (2 (c+d x))-12 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^3 e f^2 x^2 \sin (2 (c+d x))-4 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^3 f^3 x^3 \sin (2 (c+d x))}{16 b^3 \sqrt {-\left (a^2-b^2\right )^2} d^4} \]
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\[\int \frac {\left (f x +e \right )^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3008 vs. \(2 (712) = 1424\).
Time = 0.59 (sec) , antiderivative size = 3008, normalized size of antiderivative = 3.75 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sin \left (d x + c\right )^{3}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]
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