Integrand size = 25, antiderivative size = 297 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {9}{16} \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) x+\frac {3 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{20 d f}+\frac {3 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \cos (e+f x) \sin (e+f x)}{80 f}+\frac {3 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{40 d f}+\frac {3 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d f}+\frac {3 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{10 d f}-\frac {3 \cos (e+f x) (c+d \sin (e+f x))^5}{2 d f} \]
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Time = 0.32 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2842, 2832, 2813} \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 f}+\frac {1}{16} a^2 x \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right )+\frac {a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f} \]
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Rule 2813
Rule 2832
Rule 2842
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int \left (11 a^2 d-a^2 (c-12 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^4 \, dx}{6 d} \\ & = \frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (3 a^2 d (17 c+16 d)-a^2 \left (4 c^2-48 c d-55 d^2\right ) \sin (e+f x)\right ) \, dx}{30 d} \\ & = \frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^2 d \left (64 c^2+112 c d+55 d^2\right )-3 a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \sin (e+f x)\right ) \, dx}{120 d} \\ & = \frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 a^2 d \left (184 c^3+432 c^2 d+411 c d^2+128 d^3\right )-3 a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x)\right ) \, dx}{360 d} \\ & = \frac {1}{16} a^2 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) x+\frac {a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac {a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \cos (e+f x) \sin (e+f x)}{240 f}+\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f} \\ \end{align*}
Time = 1.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.82 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {3 \cos (e+f x) \left (-32 \left (15 c^4+50 c^3 d+60 c^2 d^2+36 c d^3+8 d^4\right )-\frac {30 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-15 \left (8 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) \sin (e+f x)-64 d \left (5 c^3+15 c^2 d+9 c d^2+2 d^3\right ) \sin ^2(e+f x)-10 d^2 \left (36 c^2+48 c d+11 d^2\right ) \sin ^3(e+f x)-96 d^3 (2 c+d) \sin ^4(e+f x)-40 d^4 \sin ^5(e+f x)\right )}{80 f} \]
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Time = 2.65 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(-\frac {a^{2} \left (\left (\frac {31}{16} d^{4}+c^{4}+8 c^{3} d +12 c^{2} d^{2}+8 d^{3} c \right ) \sin \left (2 f x +2 e \right )+\left (-\frac {4}{3} c^{3} d -3 d^{3} c -\frac {5}{6} d^{4}-4 c^{2} d^{2}\right ) \cos \left (3 f x +3 e \right )-\frac {3 \left (c +\frac {5 d}{6}\right ) d^{2} \left (c +\frac {d}{2}\right ) \sin \left (4 f x +4 e \right )}{4}+\frac {d^{3} \left (c +\frac {d}{2}\right ) \cos \left (5 f x +5 e \right )}{5}+\frac {d^{4} \sin \left (6 f x +6 e \right )}{48}+\left (8 c^{4}+28 c^{3} d +36 c^{2} d^{2}+22 d^{3} c +5 d^{4}\right ) \cos \left (f x +e \right )+\left (\frac {64}{15}-\frac {11 f x}{4}\right ) d^{4}+c \left (\frac {96}{5}-12 f x \right ) d^{3}+\left (-21 f x +32\right ) c^{2} d^{2}-16 \left (f x -\frac {5}{3}\right ) c^{3} d +\left (-6 f x +8\right ) c^{4}\right )}{4 f}\) | \(235\) |
parts | \(a^{2} c^{4} x -\frac {\left (4 a^{2} c \,d^{3}+2 a^{2} d^{4}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {\left (2 a^{2} c^{4}+4 a^{2} c^{3} d \right ) \cos \left (f x +e \right )}{f}+\frac {\left (6 a^{2} c^{2} d^{2}+8 a^{2} c \,d^{3}+a^{2} d^{4}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (4 a^{2} c^{3} d +12 a^{2} c^{2} d^{2}+4 a^{2} c \,d^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (a^{2} c^{4}+8 a^{2} c^{3} d +6 a^{2} c^{2} d^{2}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\) | \(311\) |
risch | \(-\frac {31 \sin \left (2 f x +2 e \right ) a^{2} d^{4}}{64 f}-\frac {a^{2} d^{3} \cos \left (5 f x +5 e \right ) c}{20 f}-\frac {2 c^{4} a^{2} \cos \left (f x +e \right )}{f}-\frac {c^{4} a^{2} \sin \left (2 f x +2 e \right )}{4 f}+\frac {11 a^{2} d^{4} x}{16}-\frac {5 a^{2} \cos \left (f x +e \right ) d^{4}}{4 f}-\frac {a^{2} d^{4} \cos \left (5 f x +5 e \right )}{40 f}+\frac {5 \sin \left (4 f x +4 e \right ) a^{2} d^{4}}{64 f}+\frac {5 a^{2} d^{4} \cos \left (3 f x +3 e \right )}{24 f}+4 a^{2} c^{3} d x +\frac {21 a^{2} c^{2} d^{2} x}{4}+3 a^{2} c \,d^{3} x -\frac {a^{2} d^{4} \sin \left (6 f x +6 e \right )}{192 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{2} c^{2} d^{2}}{16 f}+\frac {\sin \left (4 f x +4 e \right ) a^{2} c \,d^{3}}{4 f}+\frac {a^{2} d \cos \left (3 f x +3 e \right ) c^{3}}{3 f}+\frac {a^{2} d^{2} \cos \left (3 f x +3 e \right ) c^{2}}{f}+\frac {3 a^{2} d^{3} \cos \left (3 f x +3 e \right ) c}{4 f}-\frac {7 a^{2} \cos \left (f x +e \right ) c^{3} d}{f}-\frac {9 a^{2} \cos \left (f x +e \right ) c^{2} d^{2}}{f}-\frac {11 a^{2} \cos \left (f x +e \right ) d^{3} c}{2 f}-\frac {2 \sin \left (2 f x +2 e \right ) a^{2} c^{3} d}{f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} c^{2} d^{2}}{f}-\frac {2 \sin \left (2 f x +2 e \right ) a^{2} c \,d^{3}}{f}+\frac {3 a^{2} c^{4} x}{2}\) | \(456\) |
derivativedivides | \(\frac {a^{2} c^{4} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+6 a^{2} c^{2} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {4 a^{2} c \,d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-2 a^{2} c^{4} \cos \left (f x +e \right )+8 a^{2} c^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-4 a^{2} c^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+8 a^{2} c \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 a^{2} d^{4} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c^{4} \left (f x +e \right )-4 a^{2} c^{3} d \cos \left (f x +e \right )+6 a^{2} c^{2} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d^{4} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(462\) |
default | \(\frac {a^{2} c^{4} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+6 a^{2} c^{2} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {4 a^{2} c \,d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-2 a^{2} c^{4} \cos \left (f x +e \right )+8 a^{2} c^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-4 a^{2} c^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+8 a^{2} c \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 a^{2} d^{4} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c^{4} \left (f x +e \right )-4 a^{2} c^{3} d \cos \left (f x +e \right )+6 a^{2} c^{2} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d^{4} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(462\) |
norman | \(\text {Expression too large to display}\) | \(1043\) |
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Time = 0.29 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.01 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=-\frac {96 \, {\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{5} - 320 \, {\left (a^{2} c^{3} d + 3 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (24 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 84 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 11 \, a^{2} d^{4}\right )} f x + 480 \, {\left (a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, a^{2} d^{4} \cos \left (f x + e\right )^{5} - 2 \, {\left (36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 108 \, a^{2} c^{2} d^{2} + 80 \, a^{2} c d^{3} + 21 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1136 vs. \(2 (299) = 598\).
Time = 0.50 (sec) , antiderivative size = 1136, normalized size of antiderivative = 3.82 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\text {Too large to display} \]
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Time = 0.22 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.52 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} + 960 \, {\left (f x + e\right )} a^{2} c^{4} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{3} d + 1920 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} d + 3840 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d^{2} + 180 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} + 1440 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} - 256 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 240 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{3} - 128 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{4} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} - 1920 \, a^{2} c^{4} \cos \left (f x + e\right ) - 3840 \, a^{2} c^{3} d \cos \left (f x + e\right )}{960 \, f} \]
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Time = 0.34 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.51 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {a^{2} c d^{3} \cos \left (3 \, f x + 3 \, e\right )}{3 \, f} - \frac {a^{2} d^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {a^{2} d^{4} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{16} \, {\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} x + \frac {1}{8} \, {\left (8 \, a^{2} c^{4} + 24 \, a^{2} c^{2} d^{2} + 3 \, a^{2} d^{4}\right )} x - \frac {{\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{40 \, f} + \frac {{\left (8 \, a^{2} c^{3} d + 24 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{24 \, f} - \frac {{\left (8 \, a^{2} c^{4} + 12 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (4 \, a^{2} c^{3} d + 3 \, a^{2} c d^{3}\right )} \cos \left (f x + e\right )}{f} + \frac {{\left (12 \, a^{2} c^{2} d^{2} + 16 \, a^{2} c d^{3} + 3 \, a^{2} d^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (16 \, a^{2} c^{4} + 128 \, a^{2} c^{3} d + 96 \, a^{2} c^{2} d^{2} + 128 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} - \frac {{\left (6 \, a^{2} c^{2} d^{2} + a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 10.79 (sec) , antiderivative size = 865, normalized size of antiderivative = 2.91 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx=\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (24\,c^4+64\,c^3\,d+84\,c^2\,d^2+48\,c\,d^3+11\,d^4\right )}{8\,\left (3\,a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )}\right )\,\left (24\,c^4+64\,c^3\,d+84\,c^2\,d^2+48\,c\,d^3+11\,d^4\right )}{8\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (20\,a^2\,c^4+56\,a^2\,c^3\,d+48\,a^2\,c^2\,d^2+16\,a^2\,c\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (4\,a^2\,c^4+8\,d\,a^2\,c^3\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )+4\,a^2\,c^4+\frac {32\,a^2\,d^4}{15}-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,a^2\,c^4+16\,a^2\,c^3\,d+33\,a^2\,c^2\,d^2+28\,a^2\,c\,d^3+\frac {47\,a^2\,d^4}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^2\,c^4+16\,a^2\,c^3\,d+33\,a^2\,c^2\,d^2+28\,a^2\,c\,d^3+\frac {47\,a^2\,d^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,a^2\,c^4+24\,a^2\,c^3\,d+\frac {87\,a^2\,c^2\,d^2}{2}+34\,a^2\,c\,d^3+\frac {187\,a^2\,d^4}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,a^2\,c^4+24\,a^2\,c^3\,d+\frac {87\,a^2\,c^2\,d^2}{2}+34\,a^2\,c\,d^3+\frac {187\,a^2\,d^4}{24}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (40\,a^2\,c^4+144\,a^2\,c^3\,d+192\,a^2\,c^2\,d^2+128\,a^2\,c\,d^3+32\,a^2\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (20\,a^2\,c^4+72\,a^2\,c^3\,d+96\,a^2\,c^2\,d^2+\frac {288\,a^2\,c\,d^3}{5}+\frac {64\,a^2\,d^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (40\,a^2\,c^4+\frac {400\,a^2\,c^3\,d}{3}+160\,a^2\,c^2\,d^2+96\,a^2\,c\,d^3+\frac {64\,a^2\,d^4}{3}\right )+\frac {48\,a^2\,c\,d^3}{5}+\frac {40\,a^2\,c^3\,d}{3}+16\,a^2\,c^2\,d^2}{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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