Integrand size = 35, antiderivative size = 464 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {1}{16} a^2 \left (6 A \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )+B \left (16 c^3+42 c^2 d+36 c d^2+11 d^3\right )\right ) x+\frac {a^2 \left (6 A d \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right )-B \left (2 c^5-12 c^4 d+47 c^3 d^2+208 c^2 d^3+216 c d^4+64 d^5\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {a^2 \left (6 A d \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )-B \left (4 c^4-24 c^3 d+96 c^2 d^2+284 c d^3+165 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d f}+\frac {a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (2 c^3-12 c^2 d+51 c d^2+64 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {a^2 \left (6 A (c-10 d) d-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 (2 B c-6 A d-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f} \]
[Out]
Time = 0.64 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3055, 3047, 3102, 2832, 2813} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {a^2 \left (6 A d (c-10 d)-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (2 c^3-12 c^2 d+51 c d^2+64 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {1}{16} a^2 x \left (6 A \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )+B \left (16 c^3+42 c^2 d+36 c d^2+11 d^3\right )\right )+\frac {a^2 \left (6 A d \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )-B \left (4 c^4-24 c^3 d+96 c^2 d^2+284 c d^3+165 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac {a^2 \left (6 A d \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right )-B \left (2 c^5-12 c^4 d+47 c^3 d^2+208 c^2 d^3+216 c d^4+64 d^5\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {a^2 (-6 A d+2 B c-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^4}{6 d f} \]
[In]
[Out]
Rule 2813
Rule 2832
Rule 3047
Rule 3055
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^3 (a (6 A d+B (c+4 d))-a (2 B c-6 A d-7 B d) \sin (e+f x)) \, dx}{6 d} \\ & = -\frac {B \cos (e+f x) \left (a^2+a^2 \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^2 (6 A d+B (c+4 d))+\left (-a^2 (2 B c-6 A d-7 B d)+a^2 (6 A d+B (c+4 d))\right ) \sin (e+f x)-a^2 (2 B c-6 A d-7 B d) \sin ^2(e+f x)\right ) \, dx}{6 d} \\ & = \frac {a^2 (2 B c-6 A d-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \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^2 d (B c-18 A d-16 B d)-a^2 \left (6 A (c-10 d) d-B \left (2 c^2-12 c d+55 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{30 d^2} \\ & = \frac {a^2 \left (6 A (c-10 d) d-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 (2 B c-6 A d-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \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^2 d \left (6 A d (11 c+10 d)-B \left (2 c^2-52 c d-55 d^2\right )\right )-3 a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (2 c^3-12 c^2 d+51 c d^2+64 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{120 d^2} \\ & = \frac {a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (2 c^3-12 c^2 d+51 c d^2+64 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {a^2 \left (6 A (c-10 d) d-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 (2 B c-6 A d-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \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^2 d \left (6 A d \left (31 c^2+50 c d+24 d^2\right )-B \left (2 c^3-132 c^2 d-267 c d^2-128 d^3\right )\right )-3 a^2 \left (6 A d \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )-B \left (4 c^4-24 c^3 d+96 c^2 d^2+284 c d^3+165 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{360 d^2} \\ & = \frac {1}{16} a^2 \left (6 A \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )+B \left (16 c^3+42 c^2 d+36 c d^2+11 d^3\right )\right ) x+\frac {a^2 \left (6 A d \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right )-B \left (2 c^5-12 c^4 d+47 c^3 d^2+208 c^2 d^3+216 c d^4+64 d^5\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {a^2 \left (6 A d \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )-B \left (4 c^4-24 c^3 d+96 c^2 d^2+284 c d^3+165 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d f}+\frac {a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (2 c^3-12 c^2 d+51 c d^2+64 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {a^2 \left (6 A (c-10 d) d-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 (2 B c-6 A d-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{6 d f} \\ \end{align*}
Time = 2.06 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.94 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {a^2 \cos (e+f x) \left (60 \left (6 A \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )+B \left (16 c^3+42 c^2 d+36 c d^2+11 d^3\right )\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (960 A c^3+880 B c^3+2640 A c^2 d+2400 B c^2 d+2400 A c d^2+2268 B c d^2+756 A d^3+712 B d^3-16 \left (3 A d \left (5 c^2+10 c d+4 d^2\right )+B \left (5 c^3+30 c^2 d+36 c d^2+14 d^3\right )\right ) \cos (2 (e+f x))+12 d^2 (3 B c+A d+2 B d) \cos (4 (e+f x))+240 A c^3 \sin (e+f x)+480 B c^3 \sin (e+f x)+1440 A c^2 d \sin (e+f x)+1530 B c^2 d \sin (e+f x)+1530 A c d^2 \sin (e+f x)+1620 B c d^2 \sin (e+f x)+540 A d^3 \sin (e+f x)+545 B d^3 \sin (e+f x)-90 B c^2 d \sin (3 (e+f x))-90 A c d^2 \sin (3 (e+f x))-180 B c d^2 \sin (3 (e+f x))-60 A d^3 \sin (3 (e+f x))-80 B d^3 \sin (3 (e+f x))+5 B d^3 \sin (5 (e+f x))\right )\right )}{480 f \sqrt {\cos ^2(e+f x)}} \]
[In]
[Out]
Time = 2.53 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(\frac {a^{2} \left (\left (\left (-2 A -\frac {31 B}{16}\right ) d^{3}-6 d^{2} c \left (A +B \right )-6 c^{2} d \left (A +B \right )-c^{3} \left (A +2 B \right )\right ) \sin \left (2 f x +2 e \right )+\left (\left (\frac {3 A}{4}+\frac {5 B}{6}\right ) d^{3}+2 \left (\frac {9 B}{8}+A \right ) c \,d^{2}+c^{2} \left (A +2 B \right ) d +\frac {B \,c^{3}}{3}\right ) \cos \left (3 f x +3 e \right )+\frac {3 \left (\left (\frac {2 A}{3}+\frac {5 B}{6}\right ) d^{2}+c \left (A +2 B \right ) d +B \,c^{2}\right ) d \sin \left (4 f x +4 e \right )}{8}-\frac {\left (\left (A +2 B \right ) d +3 B c \right ) d^{2} \cos \left (5 f x +5 e \right )}{20}-\frac {B \,d^{3} \sin \left (6 f x +6 e \right )}{48}+\left (\left (-5 B -\frac {11 A}{2}\right ) d^{3}-18 \left (\frac {11 B}{12}+A \right ) c \,d^{2}-21 \left (A +\frac {6 B}{7}\right ) c^{2} d -8 c^{3} \left (A +\frac {7 B}{8}\right )\right ) \cos \left (f x +e \right )+\left (-\frac {64}{15} B +3 f x A +\frac {11}{4} f x B -\frac {24}{5} A \right ) d^{3}+\frac {21 c \left (f x A +\frac {6}{7} f x B -\frac {32}{21} A -\frac {48}{35} B \right ) d^{2}}{2}+12 c^{2} \left (f x A +\frac {7}{8} f x B -\frac {5}{3} A -\frac {4}{3} B \right ) d +6 c^{3} \left (f x A +\frac {2}{3} f x B -\frac {4}{3} A -\frac {10}{9} B \right )\right )}{4 f}\) | \(324\) |
parts | \(-\frac {\left (A \,a^{2} d^{3}+3 B \,a^{2} d^{2} c +2 B \,a^{2} d^{3}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {\left (2 A \,a^{2} c^{3}+3 A \,a^{2} c^{2} d +B \,a^{2} c^{3}\right ) \cos \left (f x +e \right )}{f}+\frac {\left (3 A \,a^{2} d^{2} c +2 A \,a^{2} d^{3}+3 B \,a^{2} c^{2} d +6 B \,a^{2} d^{2} c +B \,a^{2} d^{3}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {\left (A \,a^{2} c^{3}+6 A \,a^{2} c^{2} d +3 A \,a^{2} d^{2} c +2 B \,a^{2} c^{3}+3 B \,a^{2} c^{2} d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (3 A \,a^{2} c^{2} d +6 A \,a^{2} d^{2} c +A \,a^{2} d^{3}+B \,a^{2} c^{3}+6 B \,a^{2} c^{2} d +3 B \,a^{2} d^{2} c \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+a^{2} c^{3} x A +\frac {B \,a^{2} d^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\) | \(400\) |
derivativedivides | \(\frac {A \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A \,a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 A \,a^{2} d^{2} c \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 {A \,a^{2} 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}-\frac {B \,a^{2} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 B \,a^{2} 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 B \,a^{2} d^{2} c \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}+B \,a^{2} d^{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 )-2 A \,a^{2} c^{3} \cos \left (f x +e \right )+6 A \,a^{2} c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 A \,a^{2} d^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 A \,a^{2} 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 )+2 B \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 B \,a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+6 B \,a^{2} d^{2} c \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 B \,a^{2} 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 \,a^{2} c^{3} \left (f x +e \right )-3 A \,a^{2} c^{2} d \cos \left (f x +e \right )+3 A \,a^{2} d^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A \,a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,a^{2} c^{3} \cos \left (f x +e \right )+3 B \,a^{2} c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} d^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+B \,a^{2} 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 )}{f}\) | \(745\) |
default | \(\frac {A \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A \,a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 A \,a^{2} d^{2} c \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 {A \,a^{2} 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}-\frac {B \,a^{2} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 B \,a^{2} 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 B \,a^{2} d^{2} c \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}+B \,a^{2} d^{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 )-2 A \,a^{2} c^{3} \cos \left (f x +e \right )+6 A \,a^{2} c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 A \,a^{2} d^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 A \,a^{2} 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 )+2 B \,a^{2} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 B \,a^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+6 B \,a^{2} d^{2} c \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 B \,a^{2} 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 \,a^{2} c^{3} \left (f x +e \right )-3 A \,a^{2} c^{2} d \cos \left (f x +e \right )+3 A \,a^{2} d^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A \,a^{2} d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,a^{2} c^{3} \cos \left (f x +e \right )+3 B \,a^{2} c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} d^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+B \,a^{2} 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 )}{f}\) | \(745\) |
risch | \(-\frac {3 a^{2} d^{2} \cos \left (5 f x +5 e \right ) B c}{80 f}+\frac {3 \sin \left (4 f x +4 e \right ) A \,a^{2} d^{2} c}{32 f}+\frac {3 \sin \left (4 f x +4 e \right ) B \,a^{2} c^{2} d}{32 f}+\frac {3 \sin \left (4 f x +4 e \right ) B \,a^{2} d^{2} c}{16 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) c^{2} d A}{4 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) d^{2} c A}{2 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) c^{2} d B}{2 f}+\frac {9 a^{2} \cos \left (3 f x +3 e \right ) d^{2} c B}{16 f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{2} c^{2} d}{2 f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{2} d^{2} c}{2 f}-\frac {3 \sin \left (2 f x +2 e \right ) B \,a^{2} c^{2} d}{2 f}-\frac {3 \sin \left (2 f x +2 e \right ) B \,a^{2} d^{2} c}{2 f}-\frac {21 a^{2} \cos \left (f x +e \right ) c^{2} d A}{4 f}-\frac {9 a^{2} \cos \left (f x +e \right ) d^{2} c A}{2 f}-\frac {9 a^{2} \cos \left (f x +e \right ) c^{2} d B}{2 f}-\frac {33 a^{2} \cos \left (f x +e \right ) d^{2} c B}{8 f}+\frac {3 A \,a^{2} d^{3} x}{4}+B \,a^{2} c^{3} x +\frac {11 B \,a^{2} d^{3} x}{16}+3 A \,a^{2} c^{2} d x +\frac {21 A \,a^{2} c \,d^{2} x}{8}+\frac {\sin \left (4 f x +4 e \right ) A \,a^{2} d^{3}}{16 f}+\frac {5 \sin \left (4 f x +4 e \right ) B \,a^{2} d^{3}}{64 f}+\frac {3 a^{2} \cos \left (3 f x +3 e \right ) A \,d^{3}}{16 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) B \,c^{3}}{12 f}+\frac {5 a^{2} \cos \left (3 f x +3 e \right ) d^{3} B}{24 f}+\frac {3 a^{2} c^{3} x A}{2}+\frac {21 B \,a^{2} c^{2} d x}{8}+\frac {9 B \,a^{2} c \,d^{2} x}{4}-\frac {2 a^{2} \cos \left (f x +e \right ) A \,c^{3}}{f}-\frac {11 a^{2} \cos \left (f x +e \right ) A \,d^{3}}{8 f}-\frac {7 a^{2} \cos \left (f x +e \right ) B \,c^{3}}{4 f}-\frac {5 a^{2} \cos \left (f x +e \right ) d^{3} B}{4 f}-\frac {a^{2} d^{3} \cos \left (5 f x +5 e \right ) A}{80 f}-\frac {a^{2} d^{3} \cos \left (5 f x +5 e \right ) B}{40 f}-\frac {B \,a^{2} d^{3} \sin \left (6 f x +6 e \right )}{192 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} c^{3}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} d^{3}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{2} c^{3}}{2 f}-\frac {31 \sin \left (2 f x +2 e \right ) B \,a^{2} d^{3}}{64 f}\) | \(749\) |
norman | \(\text {Expression too large to display}\) | \(1554\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.78 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {48 \, {\left (3 \, B a^{2} c d^{2} + {\left (A + 2 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (B a^{2} c^{3} + 3 \, {\left (A + 2 \, B\right )} a^{2} c^{2} d + 3 \, {\left (2 \, A + 3 \, B\right )} a^{2} c d^{2} + {\left (3 \, A + 4 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (8 \, {\left (3 \, A + 2 \, B\right )} a^{2} c^{3} + 6 \, {\left (8 \, A + 7 \, B\right )} a^{2} c^{2} d + 6 \, {\left (7 \, A + 6 \, B\right )} a^{2} c d^{2} + {\left (12 \, A + 11 \, B\right )} a^{2} d^{3}\right )} f x + 480 \, {\left ({\left (A + B\right )} a^{2} c^{3} + 3 \, {\left (A + B\right )} a^{2} c^{2} d + 3 \, {\left (A + B\right )} a^{2} c d^{2} + {\left (A + B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, B a^{2} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, B a^{2} c^{2} d + 18 \, {\left (A + 2 \, B\right )} a^{2} c d^{2} + {\left (12 \, A + 19 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (A + 2 \, B\right )} a^{2} c^{3} + 6 \, {\left (8 \, A + 9 \, B\right )} a^{2} c^{2} d + 6 \, {\left (9 \, A + 10 \, B\right )} a^{2} c d^{2} + {\left (20 \, A + 21 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1865 vs. \(2 (450) = 900\).
Time = 0.52 (sec) , antiderivative size = 1865, normalized size of antiderivative = 4.02 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.56 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\frac {240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{3} + 960 \, {\left (f x + e\right )} A a^{2} c^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{3} + 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{3} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c^{2} d + 1440 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{2} d + 1920 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} 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 )} B a^{2} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{2} d + 1920 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c d^{2} + 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 a^{2} c d^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} 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 )} B a^{2} c d^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c 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 )} B a^{2} c d^{2} - 64 \, {\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 a^{2} d^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} d^{3} + 60 \, {\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 a^{2} 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 )} B a^{2} 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 )} B a^{2} d^{3} + 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 )} B a^{2} d^{3} - 1920 \, A a^{2} c^{3} \cos \left (f x + e\right ) - 960 \, B a^{2} c^{3} \cos \left (f x + e\right ) - 2880 \, A a^{2} c^{2} d \cos \left (f x + e\right )}{960 \, f} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.01 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=-\frac {B a^{2} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (24 \, A a^{2} c^{3} + 16 \, B a^{2} c^{3} + 48 \, A a^{2} c^{2} d + 42 \, B a^{2} c^{2} d + 42 \, A a^{2} c d^{2} + 36 \, B a^{2} c d^{2} + 12 \, A a^{2} d^{3} + 11 \, B a^{2} d^{3}\right )} x - \frac {{\left (3 \, B a^{2} c d^{2} + A a^{2} d^{3} + 2 \, B a^{2} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, B a^{2} c^{3} + 12 \, A a^{2} c^{2} d + 24 \, B a^{2} c^{2} d + 24 \, A a^{2} c d^{2} + 27 \, B a^{2} c d^{2} + 9 \, A a^{2} d^{3} + 10 \, B a^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, A a^{2} c^{3} + 14 \, B a^{2} c^{3} + 42 \, A a^{2} c^{2} d + 36 \, B a^{2} c^{2} d + 36 \, A a^{2} c d^{2} + 33 \, B a^{2} c d^{2} + 11 \, A a^{2} d^{3} + 10 \, B a^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (6 \, B a^{2} c^{2} d + 6 \, A a^{2} c d^{2} + 12 \, B a^{2} c d^{2} + 4 \, A a^{2} d^{3} + 5 \, B a^{2} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (16 \, A a^{2} c^{3} + 32 \, B a^{2} c^{3} + 96 \, A a^{2} c^{2} d + 96 \, B a^{2} c^{2} d + 96 \, A a^{2} c d^{2} + 96 \, B a^{2} c d^{2} + 32 \, A a^{2} d^{3} + 31 \, B a^{2} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
[In]
[Out]
Time = 15.94 (sec) , antiderivative size = 1291, normalized size of antiderivative = 2.78 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]
[In]
[Out]