Optimal. Leaf size=149 \[ \frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]
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Rubi [A] time = 0.21, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2565, 14, 2568, 2635, 8} \[ \frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2565
Rule 2568
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{10} (3 a) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{80} (3 a) \int \cos ^6(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{32} a \int \cos ^4(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{128} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{256} (3 a) \int 1 \, dx\\ &=\frac {3 a x}{256}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 101, normalized size = 0.68 \[ \frac {a (1260 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-630 \sin (6 (c+d x))+315 \sin (8 (c+d x))+126 \sin (10 (c+d x))-15120 \cos (c+d x)-6720 \cos (3 (c+d x))+1080 \cos (7 (c+d x))+280 \cos (9 (c+d x))+7560 d x)}{645120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 95, normalized size = 0.64 \[ \frac {8960 \, a \cos \left (d x + c\right )^{9} - 11520 \, a \cos \left (d x + c\right )^{7} + 945 \, a d x + 63 \, {\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 137, normalized size = 0.92 \[ \frac {3}{256} \, a x + \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} + \frac {a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 116, normalized size = 0.78 \[ \frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 76, normalized size = 0.51 \[ \frac {10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{645120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.19, size = 414, normalized size = 2.78 \[ \frac {3\,a\,x}{256}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\left (\frac {a\,\left (42525\,c+42525\,d\,x-322560\right )}{80640}-\frac {135\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {867\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\left (\frac {a\,\left (113400\,c+113400\,d\,x+215040\right )}{80640}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {519\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (198450\,c+198450\,d\,x-1075200\right )}{80640}-\frac {315\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1879\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\left (\frac {a\,\left (238140\,c+238140\,d\,x-645120\right )}{80640}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1879\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {519\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (113400\,c+113400\,d\,x-829440\right )}{80640}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {867\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\left (\frac {a\,\left (42525\,c+42525\,d\,x+92160\right )}{80640}-\frac {135\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\left (\frac {a\,\left (9450\,c+9450\,d\,x-51200\right )}{80640}-\frac {15\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {a\,\left (945\,c+945\,d\,x-5120\right )}{80640}-\frac {3\,a\,\left (c+d\,x\right )}{256}}{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 = 29.35, size = 294, normalized size = 1.97 \[ \begin {cases} \frac {3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{3}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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