Integrand size = 25, antiderivative size = 192 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {27}{16} \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) x-\frac {108 (c+d)^3 \cos (e+f x)}{f}+\frac {9 (c+d)^2 (c+7 d) \cos ^3(e+f x)}{f}-\frac {81 d^2 (c+d) \cos ^5(e+f x)}{5 f}-\frac {27 \left (24 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \cos (e+f x) \sin (e+f x)}{16 f}-\frac {9 d \left (18 c^2+54 c d+23 d^2\right ) \cos (e+f x) \sin ^3(e+f x)}{8 f}-\frac {9 d^3 \cos (e+f x) \sin ^5(e+f x)}{2 f} \]
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Time = 0.38 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.70, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2842, 3047, 3102, 2832, 2813} \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}-\frac {a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {1}{16} a^3 x \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )-\frac {a^3 \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 d f}-\frac {a^3 \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right ) \cos (e+f x)}{60 d^2 f}+\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f} \]
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Rule 2813
Rule 2832
Rule 2842
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (a+a \sin (e+f x)) \left (a^2 (c+10 d)-a^2 (2 c-13 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3 \, dx}{6 d} \\ & = -\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (a^3 (c+10 d)+\left (-a^3 (2 c-13 d)+a^3 (c+10 d)\right ) \sin (e+f x)-a^3 (2 c-13 d) \sin ^2(e+f x)\right ) \, dx}{6 d} \\ & = \frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (-3 a^3 (c-34 d) d+a^3 \left (2 c^2-18 c d+115 d^2\right ) \sin (e+f x)\right ) \, dx}{30 d^2} \\ & = -\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (-3 a^3 d \left (2 c^2-118 c d-115 d^2\right )+3 a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \sin (e+f x)\right ) \, dx}{120 d^2} \\ & = -\frac {a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x)) \left (-3 a^3 d \left (2 c^3-318 c^2 d-567 c d^2-272 d^3\right )+3 a^3 \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right ) \sin (e+f x)\right ) \, dx}{360 d^2} \\ & = \frac {1}{16} a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) x-\frac {a^3 \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right ) \cos (e+f x)}{60 d^2 f}-\frac {a^3 \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right ) \cos (e+f x) \sin (e+f x)}{240 d f}-\frac {a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.12 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {9 \cos (e+f x) \left (-16 \left (55 c^3+135 c^2 d+114 c d^2+34 d^3\right )-\frac {30 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-15 \left (24 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \sin (e+f x)-16 \left (5 c^3+45 c^2 d+57 c d^2+17 d^3\right ) \sin ^2(e+f x)-10 d \left (18 c^2+54 c d+23 d^2\right ) \sin ^3(e+f x)-144 d^2 (c+d) \sin ^4(e+f x)-40 d^3 \sin ^5(e+f x)\right )}{80 f} \]
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Time = 3.15 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.02
method | result | size |
parallelrisch | \(-\frac {3 \left (-\frac {\left (c +d \right ) \left (c^{2}+8 c d +\frac {19}{4} d^{2}\right ) \cos \left (3 f x +3 e \right )}{9}+\left (\frac {21}{16} d^{3}+c^{3}+4 c^{2} d +4 c \,d^{2}\right ) \sin \left (2 f x +2 e \right )-\frac {\left (c^{2}+3 c d +\frac {3}{2} d^{2}\right ) d \sin \left (4 f x +4 e \right )}{8}+\frac {d^{2} \left (c +d \right ) \cos \left (5 f x +5 e \right )}{20}+\frac {d^{3} \sin \left (6 f x +6 e \right )}{144}+\left (5 c^{3}+\frac {7}{2} d^{3}+13 c^{2} d +\frac {23}{2} c \,d^{2}\right ) \cos \left (f x +e \right )+\left (\frac {136}{45}-\frac {23 f x}{12}\right ) d^{3}+c \left (\frac {152}{15}-\frac {13 f x}{2}\right ) d^{2}+c^{2} \left (-\frac {15 f x}{2}+12\right ) d -\frac {10 \left (f x -\frac {22}{15}\right ) c^{3}}{3}\right ) a^{3}}{4 f}\) | \(196\) |
parts | \(a^{3} c^{3} x -\frac {\left (3 a^{3} c \,d^{2}+3 d^{3} a^{3}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {\left (3 a^{3} c^{3}+3 a^{3} c^{2} d \right ) \cos \left (f x +e \right )}{f}+\frac {\left (3 a^{3} c^{2} d +9 a^{3} c \,d^{2}+3 d^{3} a^{3}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {\left (3 a^{3} c^{3}+9 a^{3} c^{2} d +3 a^{3} c \,d^{2}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (a^{3} c^{3}+9 a^{3} c^{2} d +9 a^{3} c \,d^{2}+d^{3} a^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {d^{3} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\) | \(312\) |
risch | \(\frac {5 a^{3} c^{3} x}{2}+\frac {45 a^{3} c^{2} d x}{8}+\frac {39 a^{3} c \,d^{2} x}{8}+\frac {23 a^{3} d^{3} x}{16}-\frac {15 a^{3} \cos \left (f x +e \right ) c^{3}}{4 f}-\frac {39 a^{3} \cos \left (f x +e \right ) c^{2} d}{4 f}-\frac {69 a^{3} \cos \left (f x +e \right ) c \,d^{2}}{8 f}-\frac {21 a^{3} \cos \left (f x +e \right ) d^{3}}{8 f}-\frac {d^{3} a^{3} \sin \left (6 f x +6 e \right )}{192 f}-\frac {3 a^{3} d^{2} \cos \left (5 f x +5 e \right ) c}{80 f}-\frac {3 a^{3} d^{3} \cos \left (5 f x +5 e \right )}{80 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{3} c^{2} d}{32 f}+\frac {9 \sin \left (4 f x +4 e \right ) a^{3} c \,d^{2}}{32 f}+\frac {9 \sin \left (4 f x +4 e \right ) d^{3} a^{3}}{64 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) c^{3}}{12 f}+\frac {3 a^{3} \cos \left (3 f x +3 e \right ) c^{2} d}{4 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) c \,d^{2}}{16 f}+\frac {19 a^{3} \cos \left (3 f x +3 e \right ) d^{3}}{48 f}-\frac {3 c^{3} a^{3} \sin \left (2 f x +2 e \right )}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{3} c^{2} d}{f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{3} c \,d^{2}}{f}-\frac {63 \sin \left (2 f x +2 e \right ) d^{3} a^{3}}{64 f}\) | \(397\) |
derivativedivides | \(\frac {-\frac {a^{3} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a^{3} c^{2} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 a^{3} c \,d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+d^{3} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+3 c^{3} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+9 a^{3} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 d^{3} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-3 a^{3} c^{3} \cos \left (f x +e \right )+9 a^{3} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 d^{3} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+c^{3} a^{3} \left (f x +e \right )-3 a^{3} c^{2} d \cos \left (f x +e \right )+3 a^{3} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{3} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(481\) |
default | \(\frac {-\frac {a^{3} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a^{3} c^{2} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 a^{3} c \,d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+d^{3} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+3 c^{3} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+9 a^{3} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 d^{3} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-3 a^{3} c^{3} \cos \left (f x +e \right )+9 a^{3} c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 d^{3} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+c^{3} a^{3} \left (f x +e \right )-3 a^{3} c^{2} d \cos \left (f x +e \right )+3 a^{3} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{3} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(481\) |
norman | \(\frac {\left (\frac {5}{2} a^{3} c^{3}+\frac {45}{8} a^{3} c^{2} d +\frac {39}{8} a^{3} c \,d^{2}+\frac {23}{16} d^{3} a^{3}\right ) x +\left (15 a^{3} c^{3}+\frac {135}{4} a^{3} c^{2} d +\frac {117}{4} a^{3} c \,d^{2}+\frac {69}{8} d^{3} a^{3}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (15 a^{3} c^{3}+\frac {135}{4} a^{3} c^{2} d +\frac {117}{4} a^{3} c \,d^{2}+\frac {69}{8} d^{3} a^{3}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (50 a^{3} c^{3}+\frac {225}{2} a^{3} c^{2} d +\frac {195}{2} a^{3} c \,d^{2}+\frac {115}{4} d^{3} a^{3}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5}{2} a^{3} c^{3}+\frac {45}{8} a^{3} c^{2} d +\frac {39}{8} a^{3} c \,d^{2}+\frac {23}{16} d^{3} a^{3}\right ) x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {75}{2} a^{3} c^{3}+\frac {675}{8} a^{3} c^{2} d +\frac {585}{8} a^{3} c \,d^{2}+\frac {345}{16} d^{3} a^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {75}{2} a^{3} c^{3}+\frac {675}{8} a^{3} c^{2} d +\frac {585}{8} a^{3} c \,d^{2}+\frac {345}{16} d^{3} a^{3}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {110 a^{3} c^{3}+270 a^{3} c^{2} d +228 a^{3} c \,d^{2}+68 d^{3} a^{3}}{15 f}-\frac {6 \left (a^{3} c^{3}+a^{3} c^{2} d \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (34 a^{3} c^{3}+66 a^{3} c^{2} d +36 a^{3} c \,d^{2}+4 d^{3} a^{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {4 \left (55 a^{3} c^{3}+135 a^{3} c^{2} d +114 a^{3} c \,d^{2}+34 d^{3} a^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (76 a^{3} c^{3}+204 a^{3} c^{2} d +192 a^{3} c \,d^{2}+64 d^{3} a^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (95 a^{3} c^{3}+255 a^{3} c^{2} d +228 a^{3} c \,d^{2}+68 d^{3} a^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}-\frac {3 a^{3} \left (8 c^{3}+38 c^{2} d +50 c \,d^{2}+25 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {3 a^{3} \left (8 c^{3}+38 c^{2} d +50 c \,d^{2}+25 d^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{3} \left (24 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {a^{3} \left (24 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a^{3} \left (216 c^{3}+954 c^{2} d +1134 c \,d^{2}+391 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}+\frac {a^{3} \left (216 c^{3}+954 c^{2} d +1134 c \,d^{2}+391 d^{3}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{6}}\) | \(869\) |
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Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.36 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {144 \, {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (a^{3} c^{3} + 9 \, a^{3} c^{2} d + 15 \, a^{3} c d^{2} + 7 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} f x + 960 \, {\left (a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, a^{3} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 31 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (24 \, a^{3} c^{3} + 102 \, a^{3} c^{2} d + 114 \, a^{3} c d^{2} + 41 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1176 vs. \(2 (206) = 412\).
Time = 0.49 (sec) , antiderivative size = 1176, normalized size of antiderivative = 6.12 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (203) = 406\).
Time = 0.22 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.44 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{3} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{3} + 960 \, {\left (f x + e\right )} a^{3} c^{3} + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{2} d + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{2} d + 2160 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{2} d - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c d^{2} + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c d^{2} + 270 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d^{2} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} d^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d^{3} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{3} + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{3} - 2880 \, a^{3} c^{3} \cos \left (f x + e\right ) - 2880 \, a^{3} c^{2} d \cos \left (f x + e\right )}{960 \, f} \]
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Time = 0.56 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.90 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {a^{3} d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {a^{3} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} - \frac {3 \, a^{3} c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{16} \, {\left (24 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} x + \frac {1}{2} \, {\left (2 \, a^{3} c^{3} + 3 \, a^{3} c d^{2}\right )} x - \frac {3 \, {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, a^{3} c^{3} + 36 \, a^{3} c^{2} d + 51 \, a^{3} c d^{2} + 15 \, a^{3} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {3 \, {\left (10 \, a^{3} c^{3} + 18 \, a^{3} c^{2} d + 23 \, a^{3} c d^{2} + 5 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {3 \, {\left (4 \, a^{3} c^{2} d + a^{3} d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} + \frac {3 \, {\left (2 \, a^{3} c^{2} d + 6 \, a^{3} c d^{2} + 3 \, a^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {3 \, {\left (16 \, a^{3} c^{3} + 64 \, a^{3} c^{2} d + 48 \, a^{3} c d^{2} + 21 \, a^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
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Time = 8.68 (sec) , antiderivative size = 773, normalized size of antiderivative = 4.03 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{8\,\left (5\,a^3\,c^3+\frac {45\,a^3\,c^2\,d}{4}+\frac {39\,a^3\,c\,d^2}{4}+\frac {23\,a^3\,d^3}{8}\right )}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{8\,f}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{8\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (6\,a^3\,c^3+6\,d\,a^3\,c^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (3\,a^3\,c^3+\frac {45\,a^3\,c^2\,d}{4}+\frac {39\,a^3\,c\,d^2}{4}+\frac {23\,a^3\,d^3}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (34\,a^3\,c^3+66\,a^3\,c^2\,d+36\,a^3\,c\,d^2+4\,a^3\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (6\,a^3\,c^3+\frac {57\,a^3\,c^2\,d}{2}+\frac {75\,a^3\,c\,d^2}{2}+\frac {75\,a^3\,d^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (6\,a^3\,c^3+\frac {57\,a^3\,c^2\,d}{2}+\frac {75\,a^3\,c\,d^2}{2}+\frac {75\,a^3\,d^3}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (76\,a^3\,c^3+204\,a^3\,c^2\,d+192\,a^3\,c\,d^2+64\,a^3\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {220\,a^3\,c^3}{3}+180\,a^3\,c^2\,d+152\,a^3\,c\,d^2+\frac {136\,a^3\,d^3}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (38\,a^3\,c^3+102\,a^3\,c^2\,d+\frac {456\,a^3\,c\,d^2}{5}+\frac {136\,a^3\,d^3}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (9\,a^3\,c^3+\frac {159\,a^3\,c^2\,d}{4}+\frac {189\,a^3\,c\,d^2}{4}+\frac {391\,a^3\,d^3}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (9\,a^3\,c^3+\frac {159\,a^3\,c^2\,d}{4}+\frac {189\,a^3\,c\,d^2}{4}+\frac {391\,a^3\,d^3}{24}\right )+\frac {22\,a^3\,c^3}{3}+\frac {68\,a^3\,d^3}{15}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,a^3\,c^3+\frac {45\,a^3\,c^2\,d}{4}+\frac {39\,a^3\,c\,d^2}{4}+\frac {23\,a^3\,d^3}{8}\right )+\frac {76\,a^3\,c\,d^2}{5}+18\,a^3\,c^2\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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