Optimal. Leaf size=463 \[ \frac {1}{16} a^3 x \left (A \left (40 c^2+60 c d+26 d^2\right )+B \left (30 c^2+52 c d+23 d^2\right )\right )+\frac {a^3 \left (2 A d (2 c-11 d)-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}-\frac {a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d^2 f}-\frac {a^3 \left (2 A d \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right )-B \left (2 c^5-12 c^4 d+37 c^3 d^2-112 c^2 d^3-304 c d^4-136 d^5\right )\right ) \cos (e+f x)}{60 d^3 f}+\frac {(-6 A d+3 B c-8 B d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^3}{30 d^2 f}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3}{6 d f} \]
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Rubi [A] time = 1.13, antiderivative size = 463, 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} \[ -\frac {a^3 \left (2 A d \left (72 c^2 d^2-15 c^3 d+2 c^4+180 c d^3+76 d^4\right )-B \left (37 c^3 d^2-112 c^2 d^3-12 c^4 d+2 c^5-304 c d^4-136 d^5\right )\right ) \cos (e+f x)}{60 d^3 f}+\frac {a^3 \left (2 A d (2 c-11 d)-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (-12 c^2 d+2 c^3+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}-\frac {a^3 \left (2 A d \left (-30 c^2 d+4 c^3+146 c d^2+195 d^3\right )-B \left (76 c^2 d^2-24 c^3 d+4 c^4-236 c d^3-345 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d^2 f}+\frac {1}{16} a^3 x \left (A \left (40 c^2+60 c d+26 d^2\right )+B \left (30 c^2+52 c d+23 d^2\right )\right )+\frac {(-6 A d+3 B c-8 B d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^3}{30 d^2 f}-\frac {a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^3}{6 d f} \]
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))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx &=-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2 (a (2 B c+6 A d+3 B d)-a (3 B c-6 A d-8 B d) \sin (e+f x)) \, dx}{6 d}\\ &=-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^2 \left (3 a^2 \left (2 A d (c+8 d)-B \left (c^2-3 c d-13 d^2\right )\right )-3 a^2 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{30 d^2}\\ &=-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^3 \left (2 A d (c+8 d)-B \left (c^2-3 c d-13 d^2\right )\right )+\left (3 a^3 \left (2 A d (c+8 d)-B \left (c^2-3 c d-13 d^2\right )\right )-3 a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right )\right ) \sin (e+f x)-3 a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \sin ^2(e+f x)\right ) \, dx}{30 d^2}\\ &=\frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (c+d \sin (e+f x))^2 \left (-3 a^3 d \left (2 A (2 c-65 d) d-B \left (2 c^2-12 c d+115 d^2\right )\right )+3 a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{120 d^3}\\ &=-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}+\frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}+\frac {\int (c+d \sin (e+f x)) \left (-3 a^3 d \left (2 A d \left (2 c^2-165 c d-152 d^2\right )-B \left (2 c^3-12 c^2 d+263 c d^2+272 d^3\right )\right )+3 a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{360 d^3}\\ &=\frac {1}{16} a^3 \left (B \left (30 c^2+52 c d+23 d^2\right )+A \left (40 c^2+60 c d+26 d^2\right )\right ) x-\frac {a^3 \left (2 A d \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right )-B \left (2 c^5-12 c^4 d+37 c^3 d^2-112 c^2 d^3-304 c d^4-136 d^5\right )\right ) \cos (e+f x)}{60 d^3 f}-\frac {a^3 \left (2 A d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )-B \left (4 c^4-24 c^3 d+76 c^2 d^2-236 c d^3-345 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d^2 f}-\frac {a^3 \left (2 A d \left (2 c^2-15 c d+76 d^2\right )-B \left (2 c^3-12 c^2 d+41 c d^2-136 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^3 f}+\frac {a^3 \left (2 A (2 c-11 d) d-B \left (2 c^2-8 c d+21 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{40 d^3 f}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3}{6 d f}+\frac {(3 B c-6 A d-8 B d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{30 d^2 f}\\ \end {align*}
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Mathematica [A] time = 2.42, size = 355, normalized size = 0.77 \[ -\frac {a^3 \cos (e+f x) \left (60 \left (A \left (40 c^2+60 c d+26 d^2\right )+B \left (30 c^2+52 c d+23 d^2\right )\right ) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (-16 \left (A \left (5 c^2+30 c d+22 d^2\right )+B \left (15 c^2+44 c d+26 d^2\right )\right ) \cos (2 (e+f x))+12 d (A d+2 B c+3 B d) \cos (4 (e+f x))+720 A c^2 \sin (e+f x)+1840 A c^2+1980 A c d \sin (e+f x)-60 A c d \sin (3 (e+f x))+3360 A c d+1050 A d^2 \sin (e+f x)-90 A d^2 \sin (3 (e+f x))+1556 A d^2+990 B c^2 \sin (e+f x)-30 B c^2 \sin (3 (e+f x))+1680 B c^2+2100 B c d \sin (e+f x)-180 B c d \sin (3 (e+f x))+3112 B c d+1085 B d^2 \sin (e+f x)-140 B d^2 \sin (3 (e+f x))+5 B d^2 \sin (5 (e+f x))+1468 B d^2\right )\right )}{480 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 299, normalized size = 0.65 \[ -\frac {48 \, {\left (2 \, B a^{3} c d + {\left (A + 3 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left ({\left (A + 3 \, B\right )} a^{3} c^{2} + 2 \, {\left (3 \, A + 5 \, B\right )} a^{3} c d + {\left (5 \, A + 7 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (10 \, {\left (4 \, A + 3 \, B\right )} a^{3} c^{2} + 4 \, {\left (15 \, A + 13 \, B\right )} a^{3} c d + {\left (26 \, A + 23 \, B\right )} a^{3} d^{2}\right )} f x + 960 \, {\left ({\left (A + B\right )} a^{3} c^{2} + 2 \, {\left (A + B\right )} a^{3} c d + {\left (A + B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, B a^{3} d^{2} \cos \left (f x + e\right )^{5} - 2 \, {\left (6 \, B a^{3} c^{2} + 12 \, {\left (A + 3 \, B\right )} a^{3} c d + {\left (18 \, A + 31 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, {\left (12 \, A + 17 \, B\right )} a^{3} c^{2} + 4 \, {\left (17 \, A + 19 \, B\right )} a^{3} c d + {\left (38 \, A + 41 \, B\right )} a^{3} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 380, normalized size = 0.82 \[ -\frac {B a^{3} d^{2} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (40 \, A a^{3} c^{2} + 30 \, B a^{3} c^{2} + 60 \, A a^{3} c d + 52 \, B a^{3} c d + 26 \, A a^{3} d^{2} + 23 \, B a^{3} d^{2}\right )} x - \frac {{\left (2 \, B a^{3} c d + A a^{3} d^{2} + 3 \, B a^{3} d^{2}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, A a^{3} c^{2} + 12 \, B a^{3} c^{2} + 24 \, A a^{3} c d + 34 \, B a^{3} c d + 17 \, A a^{3} d^{2} + 19 \, B a^{3} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (30 \, A a^{3} c^{2} + 26 \, B a^{3} c^{2} + 52 \, A a^{3} c d + 46 \, B a^{3} c d + 23 \, A a^{3} d^{2} + 21 \, B a^{3} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, B a^{3} c^{2} + 4 \, A a^{3} c d + 12 \, B a^{3} c d + 6 \, A a^{3} d^{2} + 9 \, B a^{3} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (48 \, A a^{3} c^{2} + 64 \, B a^{3} c^{2} + 128 \, A a^{3} c d + 128 \, B a^{3} c d + 64 \, A a^{3} d^{2} + 63 \, B a^{3} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 725, normalized size = 1.57 \[ \frac {-\frac {a^{3} A \,c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 a^{3} A c 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 {a^{3} A \,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^{3} c^{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^{3} c d \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^{3} d^{2} \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 a^{3} A \,c^{2} \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^{3} A c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{3} A \,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 )-B \,a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+6 B \,a^{3} c 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^{3} 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}-3 a^{3} A \,c^{2} \cos \left (f x +e \right )+6 a^{3} A c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} A \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} c^{2} \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^{3} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 B \,a^{3} 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 )+a^{3} A \,c^{2} \left (f x +e \right )-2 a^{3} A c d \cos \left (f x +e \right )+a^{3} A \,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 )-B \,a^{3} c^{2} \cos \left (f x +e \right )+2 B \,a^{3} c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {B \,a^{3} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 704, normalized size = 1.52 \[ \frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{2} + 960 \, {\left (f x + e\right )} A a^{3} c^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{2} + 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^{3} c^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{2} + 1920 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c d + 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^{3} c d + 1440 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c d - 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^{3} c d + 1920 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c d + 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^{3} c d + 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c d - 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^{3} d^{2} + 960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} 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^{3} d^{2} + 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} 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^{3} d^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} d^{2} + 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^{3} 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 )} B a^{3} d^{2} - 2880 \, A a^{3} c^{2} \cos \left (f x + e\right ) - 960 \, B a^{3} c^{2} \cos \left (f x + e\right ) - 1920 \, A a^{3} c 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 = 15.63, size = 976, normalized size = 2.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.85, size = 1804, normalized size = 3.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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