Optimal. Leaf size=185 \[ -\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {9 a^2 x}{256} \]
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Rubi [A] time = 0.34, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2873, 2568, 2635, 8, 2565, 270} \[ -\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {9 a^2 x}{256} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^4(c+d x) \sin ^4(c+d x)+2 a^2 \cos ^4(c+d x) \sin ^5(c+d x)+a^2 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{8} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{2} a^2 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{16} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{32} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{64} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{128} \left (3 a^2\right ) \int 1 \, dx+\frac {1}{128} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^2 x}{128}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{256} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac {9 a^2 x}{256}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 116, normalized size = 0.63 \[ \frac {a^2 (-1260 \sin (2 (c+d x))-7560 \sin (4 (c+d x))+630 \sin (6 (c+d x))+945 \sin (8 (c+d x))-126 \sin (10 (c+d x))-30240 \cos (c+d x)-6720 \cos (3 (c+d x))+4032 \cos (5 (c+d x))+720 \cos (7 (c+d x))-560 \cos (9 (c+d x))+22680 c+22680 d x)}{645120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 124, normalized size = 0.67 \[ -\frac {17920 \, a^{2} \cos \left (d x + c\right )^{9} - 46080 \, a^{2} \cos \left (d x + c\right )^{7} + 32256 \, a^{2} \cos \left (d x + c\right )^{5} - 2835 \, a^{2} d x + 63 \, {\left (128 \, a^{2} \cos \left (d x + c\right )^{9} - 496 \, a^{2} \cos \left (d x + c\right )^{7} + 488 \, a^{2} \cos \left (d x + c\right )^{5} - 30 \, a^{2} \cos \left (d x + c\right )^{3} - 45 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 174, normalized size = 0.94 \[ \frac {9}{256} \, a^{2} x - \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a^{2} \cos \left (d x + c\right )}{64 \, d} - \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {3 \, a^{2} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {3 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 218, normalized size = 1.18 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 123, normalized size = 0.66 \[ -\frac {4096 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{2} + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} - 630 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{645120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.21, size = 469, normalized size = 2.54 \[ \frac {9\,a^2\,x}{256}-\frac {\frac {9\,a^2\,\left (c+d\,x\right )}{256}+\frac {87\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {553\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {491\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {2555\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {2555\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {491\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {553\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {87\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}-\frac {a^2\,\left (2835\,c+2835\,d\,x-8192\right )}{80640}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {45\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (28350\,c+28350\,d\,x-81920\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {405\,a^2\,\left (c+d\,x\right )}{256}-\frac {a^2\,\left (127575\,c+127575\,d\,x-368640\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {135\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (340200\,c+340200\,d\,x+737280\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {945\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (595350\,c+595350\,d\,x+860160\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {135\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (340200\,c+340200\,d\,x-1720320\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {567\,a^2\,\left (c+d\,x\right )}{64}-\frac {a^2\,\left (714420\,c+714420\,d\,x-1032192\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {945\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (595350\,c+595350\,d\,x-2580480\right )}{80640}\right )+\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 34.60, size = 554, normalized size = 2.99 \[ \begin {cases} \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {3 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {9 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {11 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {2 a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {11 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {8 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {16 a^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \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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