Optimal. Leaf size=165 \[ \frac {2 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {13 a^2 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {13 a^2 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {13 a^2 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {13 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {13 a^2 x}{256} \]
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Rubi [A] time = 0.30, antiderivative size = 165, 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, 14} \[ \frac {2 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {13 a^2 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {13 a^2 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {13 a^2 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {13 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {13 a^2 x}{256} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^6(c+d x) \sin ^2(c+d x)+2 a^2 \cos ^6(c+d x) \sin ^3(c+d x)+a^2 \cos ^6(c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{8} a^2 \int \cos ^6(c+d x) \, dx+\frac {1}{10} \left (3 a^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {13 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{80} \left (3 a^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{48} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {13 a^2 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {13 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{32} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{64} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {13 a^2 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {13 a^2 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {13 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{128} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{128} \left (5 a^2\right ) \int 1 \, dx\\ &=\frac {5 a^2 x}{128}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {13 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {13 a^2 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {13 a^2 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {13 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{256} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac {13 a^2 x}{256}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {13 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {13 a^2 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {13 a^2 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {13 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 106, normalized size = 0.64 \[ \frac {a^2 (11340 \sin (2 (c+d x))-7560 \sin (4 (c+d x))-3990 \sin (6 (c+d x))-315 \sin (8 (c+d x))+126 \sin (10 (c+d x))-30240 \cos (c+d x)-13440 \cos (3 (c+d x))+2160 \cos (7 (c+d x))+560 \cos (9 (c+d x))+12600 c+32760 d x)}{645120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 111, normalized size = 0.67 \[ \frac {17920 \, a^{2} \cos \left (d x + c\right )^{9} - 23040 \, a^{2} \cos \left (d x + c\right )^{7} + 4095 \, a^{2} d x + 21 \, {\left (384 \, a^{2} \cos \left (d x + c\right )^{9} - 1008 \, a^{2} \cos \left (d x + c\right )^{7} + 104 \, a^{2} \cos \left (d x + c\right )^{5} + 130 \, a^{2} \cos \left (d x + c\right )^{3} + 195 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 157, normalized size = 0.95 \[ \frac {13}{256} \, a^{2} x + \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac {3 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {3 \, a^{2} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {19 \, a^{2} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {3 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {9 \, a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 184, normalized size = 1.12 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 128, normalized size = 0.78 \[ \frac {20480 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 210 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{645120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.12, size = 469, normalized size = 2.84 \[ \frac {13\,a^2\,x}{256}-\frac {\frac {13\,a^2\,\left (c+d\,x\right )}{256}-\frac {647\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384}-\frac {2311\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}+\frac {457\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {2169\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {2169\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}-\frac {457\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {2311\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{480}+\frac {647\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{384}-\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}-\frac {a^2\,\left (4095\,c+4095\,d\,x-10240\right )}{80640}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {65\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (40950\,c+40950\,d\,x-102400\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {585\,a^2\,\left (c+d\,x\right )}{256}-\frac {a^2\,\left (184275\,c+184275\,d\,x+184320\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {585\,a^2\,\left (c+d\,x\right )}{256}-\frac {a^2\,\left (184275\,c+184275\,d\,x-645120\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {195\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (491400\,c+491400\,d\,x+430080\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {195\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (491400\,c+491400\,d\,x-1658880\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {819\,a^2\,\left (c+d\,x\right )}{64}-\frac {a^2\,\left (1031940\,c+1031940\,d\,x-1290240\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {1365\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (859950\,c+859950\,d\,x-2150400\right )}{80640}\right )+\frac {13\,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 = 27.32, size = 529, normalized size = 3.21 \[ \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 {5 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 {5 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 {15 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 {5 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 {5 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 {5 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 {55 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {73 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {2 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {4 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin ^{2}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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