Optimal. Leaf size=400 \[ \frac {1}{16} \left (18 a^2 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+8 a^3 \left (2 c^3+3 c d^2\right )\right ) x-\frac {\left (3 a b^2 d \left (3 c^2+d^2\right )+3 a^2 b c \left (c^2+3 d^2\right )+b^3 c \left (c^2+3 d^2\right )+a^3 \left (3 c^2 d+d^3\right )\right ) \cos (e+f x)}{f}+\frac {(b c+a d) \left (8 a b c d+a^2 d^2+b^2 \left (c^2+6 d^2\right )\right ) \cos ^3(e+f x)}{3 f}-\frac {3 b^2 d^2 (b c+a d) \cos ^5(e+f x)}{5 f}-\frac {\left (24 a^3 c d^2+18 a^2 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{16 f}-\frac {5 b^3 d^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac {3 b d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) \cos (e+f x) \sin ^3(e+f x)}{4 f}-\frac {b^3 d^3 \cos (e+f x) \sin ^5(e+f x)}{6 f} \]
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Rubi [A]
time = 0.66, antiderivative size = 493, normalized size of antiderivative = 1.23, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2872, 3102,
2832, 2813} \begin {gather*} \frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {1}{16} x \left (8 a^3 \left (2 c^3+3 c d^2\right )+18 a^2 b d \left (4 c^2+d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+b^3 d \left (18 c^2+5 d^2\right )\right )-\frac {\left (40 a^3 d^3+90 a^2 b c d^2-a b^2 \left (18 c^2 d-96 d^3\right )+b^3 \left (2 c^3+21 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac {\left (200 a^3 c d^3+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 d \left (6 c^3-71 c d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d f}-\frac {\left (40 a^3 d^3 \left (4 c^2+d^2\right )+90 a^2 b c d^2 \left (c^2+4 d^2\right )-6 a b^2 d \left (3 c^4-52 c^2 d^2-16 d^4\right )+b^3 \left (2 c^5+17 c^3 d^2+96 c d^4\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2813
Rule 2832
Rule 2872
Rule 3102
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx &=-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (b^3 c+6 a^3 d+4 a b^2 d-b \left (a b c-18 a^2 d-5 b^2 d\right ) \sin (e+f x)-b^2 (2 b c-13 a d) \sin ^2(e+f x)\right ) \, dx}{6 d}\\ &=\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (-3 d \left (b^3 c-10 a^3 d-24 a b^2 d\right )-b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{30 d^2}\\ &=\frac {b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 d \left (40 a^3 c d+78 a b^2 c d+90 a^2 b d^2-b^3 \left (2 c^2-25 d^2\right )\right )+3 \left (90 a^2 b c d^2+40 a^3 d^3+b^3 \left (2 c^3+21 c d^2\right )-a b^2 \left (18 c^2 d-96 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{120 d^2}\\ &=-\frac {\left (90 a^2 b c d^2+40 a^3 d^3+b^3 \left (2 c^3+21 c d^2\right )-a b^2 \left (18 c^2 d-96 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 d \left (450 a^2 b c d^2+40 a^3 d \left (3 c^2+2 d^2\right )+6 a b^2 d \left (33 c^2+32 d^2\right )-b^3 \left (2 c^3-117 c d^2\right )\right )+3 \left (200 a^3 c d^3+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 d \left (6 c^3-71 c d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{360 d^2}\\ &=\frac {1}{16} \left (18 a^2 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+8 a^3 \left (2 c^3+3 c d^2\right )\right ) x-\frac {\left (40 a^3 d^3 \left (4 c^2+d^2\right )+90 a^2 b c d^2 \left (c^2+4 d^2\right )-6 a b^2 d \left (3 c^4-52 c^2 d^2-16 d^4\right )+b^3 \left (2 c^5+17 c^3 d^2+96 c d^4\right )\right ) \cos (e+f x)}{60 d^2 f}-\frac {\left (200 a^3 c d^3+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 d \left (6 c^3-71 c d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d f}-\frac {\left (90 a^2 b c d^2+40 a^3 d^3+b^3 \left (2 c^3+21 c d^2\right )-a b^2 \left (18 c^2 d-96 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\\ \end {align*}
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Mathematica [A]
time = 1.20, size = 552, normalized size = 1.38 \begin {gather*} \frac {960 a^3 c^3 e+1440 a b^2 c^3 e+4320 a^2 b c^2 d e+1080 b^3 c^2 d e+1440 a^3 c d^2 e+3240 a b^2 c d^2 e+1080 a^2 b d^3 e+300 b^3 d^3 e+960 a^3 c^3 f x+1440 a b^2 c^3 f x+4320 a^2 b c^2 d f x+1080 b^3 c^2 d f x+1440 a^3 c d^2 f x+3240 a b^2 c d^2 f x+1080 a^2 b d^3 f x+300 b^3 d^3 f x-360 \left (b^3 c \left (2 c^2+5 d^2\right )+a b^2 d \left (18 c^2+5 d^2\right )+2 a^2 b c \left (4 c^2+9 d^2\right )+2 a^3 \left (4 c^2 d+d^3\right )\right ) \cos (e+f x)+20 \left (36 a^2 b c d^2+4 a^3 d^3+3 a b^2 d \left (12 c^2+5 d^2\right )+b^3 \left (4 c^3+15 c d^2\right )\right ) \cos (3 (e+f x))-36 b^3 c d^2 \cos (5 (e+f x))-36 a b^2 d^3 \cos (5 (e+f x))-720 a b^2 c^3 \sin (2 (e+f x))-2160 a^2 b c^2 d \sin (2 (e+f x))-720 b^3 c^2 d \sin (2 (e+f x))-720 a^3 c d^2 \sin (2 (e+f x))-2160 a b^2 c d^2 \sin (2 (e+f x))-720 a^2 b d^3 \sin (2 (e+f x))-225 b^3 d^3 \sin (2 (e+f x))+90 b^3 c^2 d \sin (4 (e+f x))+270 a b^2 c d^2 \sin (4 (e+f x))+90 a^2 b d^3 \sin (4 (e+f x))+45 b^3 d^3 \sin (4 (e+f x))-5 b^3 d^3 \sin (6 (e+f x))}{960 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 489, normalized size = 1.22 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 513, normalized size = 1.28 \begin {gather*} \frac {960 \, {\left (f x + e\right )} a^{3} c^{3} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} c^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3} c^{3} + 2160 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b c^{2} d + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b^{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^{3} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, 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^{2} b 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 b^{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^{3} c d^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + 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^{2} b d^{3} - 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 b^{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^{3} d^{3} - 2880 \, a^{2} b c^{3} \cos \left (f x + e\right ) - 2880 \, a^{3} c^{2} d \cos \left (f x + e\right )}{960 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 382, normalized size = 0.96 \begin {gather*} -\frac {144 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} + {\left (a^{3} + 6 \, a b^{2}\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} + 18 \, {\left (4 \, a^{2} b + b^{3}\right )} c^{2} d + 6 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c d^{2} + {\left (18 \, a^{2} b + 5 \, b^{3}\right )} d^{3}\right )} f x + 240 \, {\left ({\left (3 \, a^{2} b + b^{3}\right )} c^{3} + 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d + 3 \, {\left (3 \, a^{2} b + b^{3}\right )} c d^{2} + {\left (a^{3} + 3 \, a b^{2}\right )} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, b^{3} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, b^{3} c^{2} d + 54 \, a b^{2} c d^{2} + {\left (18 \, a^{2} b + 13 \, b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (24 \, a b^{2} c^{3} + 6 \, {\left (12 \, a^{2} b + 5 \, b^{3}\right )} c^{2} d + 6 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} c d^{2} + {\left (30 \, a^{2} b + 11 \, b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1217 vs.
\(2 (400) = 800\).
time = 0.63, size = 1217, normalized size = 3.04 \begin {gather*} \begin {cases} a^{3} c^{3} x - \frac {3 a^{3} c^{2} d \cos {\left (e + f x \right )}}{f} + \frac {3 a^{3} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{3} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a^{3} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {a^{3} d^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{3} d^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 a^{2} b c^{3} \cos {\left (e + f x \right )}}{f} + \frac {9 a^{2} b c^{2} d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {9 a^{2} b c^{2} d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {9 a^{2} b c^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {9 a^{2} b c d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {6 a^{2} b c d^{2} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {9 a^{2} b d^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{2} b d^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {9 a^{2} b d^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {15 a^{2} b d^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {9 a^{2} b d^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {3 a b^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a b^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a b^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {9 a b^{2} c^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {6 a b^{2} c^{2} d \cos ^{3}{\left (e + f x \right )}}{f} + \frac {27 a b^{2} c d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {27 a b^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {27 a b^{2} c d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {45 a b^{2} c d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {27 a b^{2} c d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 a b^{2} d^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a b^{2} d^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 a b^{2} d^{3} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {b^{3} c^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 b^{3} c^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {9 b^{3} c^{2} d x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 b^{3} c^{2} d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {9 b^{3} c^{2} d x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {15 b^{3} c^{2} d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {9 b^{3} c^{2} d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 b^{3} c d^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 b^{3} c d^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 b^{3} c d^{2} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {5 b^{3} d^{3} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 b^{3} d^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {15 b^{3} d^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {5 b^{3} d^{3} x \cos ^{6}{\left (e + f x \right )}}{16} - \frac {11 b^{3} d^{3} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {5 b^{3} d^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {5 b^{3} d^{3} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\left (e \right )}\right )^{3} \left (c + d \sin {\left (e \right )}\right )^{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.72, size = 416, normalized size = 1.04 \begin {gather*} -\frac {b^{3} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (16 \, a^{3} c^{3} + 24 \, a b^{2} c^{3} + 72 \, a^{2} b c^{2} d + 18 \, b^{3} c^{2} d + 24 \, a^{3} c d^{2} + 54 \, a b^{2} c d^{2} + 18 \, a^{2} b d^{3} + 5 \, b^{3} d^{3}\right )} x - \frac {3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, b^{3} c^{3} + 36 \, a b^{2} c^{2} d + 36 \, a^{2} b c d^{2} + 15 \, b^{3} c d^{2} + 4 \, a^{3} d^{3} + 15 \, a b^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {3 \, {\left (8 \, a^{2} b c^{3} + 2 \, b^{3} c^{3} + 8 \, a^{3} c^{2} d + 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} + 5 \, b^{3} c d^{2} + 2 \, a^{3} d^{3} + 5 \, a b^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {3 \, {\left (2 \, b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3} + b^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {3 \, {\left (16 \, a b^{2} c^{3} + 48 \, a^{2} b c^{2} d + 16 \, b^{3} c^{2} d + 16 \, a^{3} c d^{2} + 48 \, a b^{2} c d^{2} + 16 \, a^{2} b d^{3} + 5 \, b^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.92, size = 574, normalized size = 1.44 \begin {gather*} -\frac {180\,a^3\,d^3\,\cos \left (e+f\,x\right )+180\,b^3\,c^3\,\cos \left (e+f\,x\right )-20\,a^3\,d^3\,\cos \left (3\,e+3\,f\,x\right )-20\,b^3\,c^3\,\cos \left (3\,e+3\,f\,x\right )+\frac {225\,b^3\,d^3\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {45\,b^3\,d^3\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {5\,b^3\,d^3\,\sin \left (6\,e+6\,f\,x\right )}{4}-75\,a\,b^2\,d^3\,\cos \left (3\,e+3\,f\,x\right )+9\,a\,b^2\,d^3\,\cos \left (5\,e+5\,f\,x\right )-75\,b^3\,c\,d^2\,\cos \left (3\,e+3\,f\,x\right )+9\,b^3\,c\,d^2\,\cos \left (5\,e+5\,f\,x\right )+180\,a\,b^2\,c^3\,\sin \left (2\,e+2\,f\,x\right )+180\,a^2\,b\,d^3\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,a^2\,b\,d^3\,\sin \left (4\,e+4\,f\,x\right )}{2}+180\,a^3\,c\,d^2\,\sin \left (2\,e+2\,f\,x\right )+180\,b^3\,c^2\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,b^3\,c^2\,d\,\sin \left (4\,e+4\,f\,x\right )}{2}+720\,a^2\,b\,c^3\,\cos \left (e+f\,x\right )+450\,a\,b^2\,d^3\,\cos \left (e+f\,x\right )+720\,a^3\,c^2\,d\,\cos \left (e+f\,x\right )+450\,b^3\,c\,d^2\,\cos \left (e+f\,x\right )-240\,a^3\,c^3\,f\,x-75\,b^3\,d^3\,f\,x+1620\,a\,b^2\,c^2\,d\,\cos \left (e+f\,x\right )+1620\,a^2\,b\,c\,d^2\,\cos \left (e+f\,x\right )-360\,a\,b^2\,c^3\,f\,x-270\,a^2\,b\,d^3\,f\,x-360\,a^3\,c\,d^2\,f\,x-270\,b^3\,c^2\,d\,f\,x-180\,a\,b^2\,c^2\,d\,\cos \left (3\,e+3\,f\,x\right )-180\,a^2\,b\,c\,d^2\,\cos \left (3\,e+3\,f\,x\right )+540\,a\,b^2\,c\,d^2\,\sin \left (2\,e+2\,f\,x\right )+540\,a^2\,b\,c^2\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {135\,a\,b^2\,c\,d^2\,\sin \left (4\,e+4\,f\,x\right )}{2}-810\,a\,b^2\,c\,d^2\,f\,x-1080\,a^2\,b\,c^2\,d\,f\,x}{240\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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