Optimal. Leaf size=464 \[ \frac {a^2 \left (6 A d (c-10 d)-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (2 c^3-12 c^2 d+51 c d^2+64 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {1}{16} a^2 x \left (6 A \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )+B \left (16 c^3+42 c^2 d+36 c d^2+11 d^3\right )\right )+\frac {a^2 \left (6 A d \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right )-B \left (4 c^4-24 c^3 d+96 c^2 d^2+284 c d^3+165 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac {a^2 \left (6 A d \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right )-B \left (2 c^5-12 c^4 d+47 c^3 d^2+208 c^2 d^3+216 c d^4+64 d^5\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {a^2 (-6 A d+2 B c-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^4}{6 d f} \]
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Rubi [A] time = 0.95, antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2976, 2968, 3023, 2753, 2734} \[ \frac {a^2 \left (6 A d \left (-44 c^2 d^2-10 c^3 d+c^4-40 c d^3-12 d^4\right )-B \left (47 c^3 d^2+208 c^2 d^3-12 c^4 d+2 c^5+216 c d^4+64 d^5\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {a^2 \left (6 A d (c-10 d)-B \left (2 c^2-12 c d+55 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {a^2 \left (6 A d \left (c^2-10 c d-12 d^2\right )-B \left (-12 c^2 d+2 c^3+51 c d^2+64 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {a^2 \left (6 A d \left (-20 c^2 d+2 c^3-57 c d^2-30 d^3\right )-B \left (96 c^2 d^2-24 c^3 d+4 c^4+284 c d^3+165 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac {1}{16} a^2 x \left (6 A \left (8 c^2 d+4 c^3+7 c d^2+2 d^3\right )+B \left (42 c^2 d+16 c^3+36 c d^2+11 d^3\right )\right )+\frac {a^2 (-6 A d+2 B c-7 B d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^4}{6 d f} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 2968
Rule 2976
Rule 3023
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx &=-\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*}
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Mathematica [A] time = 3.13, size = 437, normalized size = 0.94 \[ -\frac {a^2 \cos (e+f x) \left (60 \left (6 A \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right )+B \left (16 c^3+42 c^2 d+36 c d^2+11 d^3\right )\right ) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (-16 \left (3 A d \left (5 c^2+10 c d+4 d^2\right )+B \left (5 c^3+30 c^2 d+36 c d^2+14 d^3\right )\right ) \cos (2 (e+f x))+12 d^2 (A d+3 B c+2 B d) \cos (4 (e+f x))+240 A c^3 \sin (e+f x)+960 A c^3+1440 A c^2 d \sin (e+f x)+2640 A c^2 d+1530 A c d^2 \sin (e+f x)-90 A c d^2 \sin (3 (e+f x))+2400 A c d^2+540 A d^3 \sin (e+f x)-60 A d^3 \sin (3 (e+f x))+756 A d^3+480 B c^3 \sin (e+f x)+880 B c^3+1530 B c^2 d \sin (e+f x)-90 B c^2 d \sin (3 (e+f x))+2400 B c^2 d+1620 B c d^2 \sin (e+f x)-180 B c d^2 \sin (3 (e+f x))+2268 B c d^2+545 B d^3 \sin (e+f x)-80 B d^3 \sin (3 (e+f x))+5 B d^3 \sin (5 (e+f x))+712 B d^3\right )\right )}{480 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 364, normalized size = 0.78 \[ -\frac {48 \, {\left (3 \, B a^{2} c d^{2} + {\left (A + 2 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (B a^{2} c^{3} + 3 \, {\left (A + 2 \, B\right )} a^{2} c^{2} d + 3 \, {\left (2 \, A + 3 \, B\right )} a^{2} c d^{2} + {\left (3 \, A + 4 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (8 \, {\left (3 \, A + 2 \, B\right )} a^{2} c^{3} + 6 \, {\left (8 \, A + 7 \, B\right )} a^{2} c^{2} d + 6 \, {\left (7 \, A + 6 \, B\right )} a^{2} c d^{2} + {\left (12 \, A + 11 \, B\right )} a^{2} d^{3}\right )} f x + 480 \, {\left ({\left (A + B\right )} a^{2} c^{3} + 3 \, {\left (A + B\right )} a^{2} c^{2} d + 3 \, {\left (A + B\right )} a^{2} c d^{2} + {\left (A + B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, B a^{2} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, B a^{2} c^{2} d + 18 \, {\left (A + 2 \, B\right )} a^{2} c d^{2} + {\left (12 \, A + 19 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (A + 2 \, B\right )} a^{2} c^{3} + 6 \, {\left (8 \, A + 9 \, B\right )} a^{2} c^{2} d + 6 \, {\left (9 \, A + 10 \, B\right )} a^{2} c d^{2} + {\left (20 \, A + 21 \, B\right )} a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 474, normalized size = 1.02 \[ -\frac {B a^{2} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (24 \, A a^{2} c^{3} + 16 \, B a^{2} c^{3} + 48 \, A a^{2} c^{2} d + 42 \, B a^{2} c^{2} d + 42 \, A a^{2} c d^{2} + 36 \, B a^{2} c d^{2} + 12 \, A a^{2} d^{3} + 11 \, B a^{2} d^{3}\right )} x - \frac {{\left (3 \, B a^{2} c d^{2} + A a^{2} d^{3} + 2 \, B a^{2} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, B a^{2} c^{3} + 12 \, A a^{2} c^{2} d + 24 \, B a^{2} c^{2} d + 24 \, A a^{2} c d^{2} + 27 \, B a^{2} c d^{2} + 9 \, A a^{2} d^{3} + 10 \, B a^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, A a^{2} c^{3} + 14 \, B a^{2} c^{3} + 42 \, A a^{2} c^{2} d + 36 \, B a^{2} c^{2} d + 36 \, A a^{2} c d^{2} + 33 \, B a^{2} c d^{2} + 11 \, A a^{2} d^{3} + 10 \, B a^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (6 \, B a^{2} c^{2} d + 6 \, A a^{2} c d^{2} + 12 \, B a^{2} c d^{2} + 4 \, A a^{2} d^{3} + 5 \, B a^{2} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (16 \, A a^{2} c^{3} + 32 \, B a^{2} c^{3} + 96 \, A a^{2} c^{2} d + 96 \, B a^{2} c^{2} d + 96 \, A a^{2} c d^{2} + 96 \, B a^{2} c d^{2} + 32 \, A a^{2} d^{3} + 31 \, B a^{2} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 745, normalized size = 1.61 \[ \frac {a^{2} A \,c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} A \,c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{2} A 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 {a^{2} A \,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} 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}+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^{2} A \,c^{3} \cos \left (f x +e \right )+6 a^{2} A \,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 )-2 a^{2} A c \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+2 a^{2} A \,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 {\sin \left (f x +e \right ) \cos \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} 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 {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^{2} A \,c^{3} \left (f x +e \right )-3 a^{2} A \,c^{2} d \cos \left (f x +e \right )+3 a^{2} A 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 {a^{2} A \,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 {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \,d^{2} \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 724, normalized size = 1.56 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.99, size = 1291, normalized size = 2.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.95, size = 1865, normalized size = 4.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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