Optimal. Leaf size=143 \[ -\frac {a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a x}{128}-\frac {b \cos ^9(c+d x)}{9 d}+\frac {2 b \cos ^7(c+d x)}{7 d}-\frac {b \cos ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.19, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2568, 2635, 8, 2565, 270} \[ -\frac {a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a x}{128}-\frac {b \cos ^9(c+d x)}{9 d}+\frac {2 b \cos ^7(c+d x)}{7 d}-\frac {b \cos ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+b \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{8} (3 a) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {b \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{16} a \int \cos ^4(c+d x) \, dx-\frac {b \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {b \cos ^5(c+d x)}{5 d}+\frac {2 b \cos ^7(c+d x)}{7 d}-\frac {b \cos ^9(c+d x)}{9 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{64} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {b \cos ^5(c+d x)}{5 d}+\frac {2 b \cos ^7(c+d x)}{7 d}-\frac {b \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{128} (3 a) \int 1 \, dx\\ &=\frac {3 a x}{128}-\frac {b \cos ^5(c+d x)}{5 d}+\frac {2 b \cos ^7(c+d x)}{7 d}-\frac {b \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 92, normalized size = 0.64 \[ \frac {-2520 a \sin (4 (c+d x))+315 a \sin (8 (c+d x))+7560 a c+7560 a d x-7560 b \cos (c+d x)-1680 b \cos (3 (c+d x))+1008 b \cos (5 (c+d x))+180 b \cos (7 (c+d x))-140 b \cos (9 (c+d x))}{322560 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 95, normalized size = 0.66 \[ -\frac {4480 \, b \cos \left (d x + c\right )^{9} - 11520 \, b \cos \left (d x + c\right )^{7} + 8064 \, b \cos \left (d x + c\right )^{5} - 945 \, a d x - 315 \, {\left (16 \, a \cos \left (d x + c\right )^{7} - 24 \, a \cos \left (d x + c\right )^{5} + 2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 107, normalized size = 0.75 \[ \frac {3}{128} \, a x - \frac {b \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {b \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {b \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {b \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, b \cos \left (d x + c\right )}{128 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 124, normalized size = 0.87 \[ \frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 71, normalized size = 0.50 \[ \frac {315 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a - 1024 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b}{322560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.32, size = 223, normalized size = 1.56 \[ \frac {3\,a\,x}{128}-\frac {-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}+\frac {155\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {32\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {169\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}-16\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {112\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}+\frac {169\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {32\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}-\frac {155\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\frac {64\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {16\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {16\,b}{315}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.15, size = 272, normalized size = 1.90 \[ \begin {cases} \frac {3 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {11 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {b \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right ) \sin ^{4}{\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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