Optimal. Leaf size=183 \[ -\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {19 a^3 x}{256} \]
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Rubi [A] time = 0.33, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2568, 2635, 8, 2565, 14, 270} \[ -\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {19 a^3 x}{256} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 270
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))^3 \, dx &=\int \left (a^3 \cos ^6(c+d x) \sin ^2(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^4(c+d x)+a^3 \cos ^6(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{8} a^3 \int \cos ^6(c+d x) \, dx+\frac {1}{10} \left (9 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{48} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{80} \left (9 a^3\right ) \int \cos ^6(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{64} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{128} \left (5 a^3\right ) \int 1 \, dx+\frac {1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {5 a^3 x}{128}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{256} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac {19 a^3 x}{256}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 1.14, size = 126, normalized size = 0.69 \[ \frac {a^3 (152460 \sin (2 (c+d x))-138600 \sin (4 (c+d x))-57750 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+4158 \sin (10 (c+d x))-568260 \cos (c+d x)-244860 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+40590 \cos (7 (c+d x))+8470 \cos (9 (c+d x))-630 \cos (11 (c+d x))+415800 c+526680 d x)}{7096320 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 124, normalized size = 0.68 \[ -\frac {80640 \, a^{3} \cos \left (d x + c\right )^{11} - 492800 \, a^{3} \cos \left (d x + c\right )^{9} + 506880 \, a^{3} \cos \left (d x + c\right )^{7} - 65835 \, a^{3} d x - 231 \, {\left (1152 \, a^{3} \cos \left (d x + c\right )^{9} - 2064 \, a^{3} \cos \left (d x + c\right )^{7} + 152 \, a^{3} \cos \left (d x + c\right )^{5} + 190 \, a^{3} \cos \left (d x + c\right )^{3} + 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 191, normalized size = 1.04 \[ \frac {19}{256} \, a^{3} x - \frac {a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {11 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {41 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {53 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {41 \, a^{3} \cos \left (d x + c\right )}{512 \, d} + \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {25 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {11 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 236, normalized size = 1.29 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 164, normalized size = 0.90 \[ -\frac {10240 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 337920 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 2079 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 2310 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{7096320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.12, size = 543, normalized size = 2.97 \[ \frac {19\,a^3\,x}{256}-\frac {\frac {19\,a^3\,\left (c+d\,x\right )}{256}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}-\frac {32417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1920}+\frac {466\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}-\frac {2937\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {2937\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}-\frac {466\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{15}+\frac {32417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{1920}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{12}-\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}-a^3\,\left (\frac {19\,c}{256}+\frac {19\,d\,x}{256}-\frac {148}{693}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {209\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {209\,c}{256}+\frac {209\,d\,x}{256}-\frac {148}{63}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {1045\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {1045\,c}{256}+\frac {1045\,d\,x}{256}-12\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {1045\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {1045\,c}{256}+\frac {1045\,d\,x}{256}+\frac {16}{63}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {3135\,c}{128}+\frac {3135\,d\,x}{128}-\frac {16}{3}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {3135\,c}{256}+\frac {3135\,d\,x}{256}-\frac {44}{3}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {3135\,c}{128}+\frac {3135\,d\,x}{128}-\frac {456}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {3135\,c}{256}+\frac {3135\,d\,x}{256}-\frac {144}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {4389\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {4389\,c}{128}+\frac {4389\,d\,x}{128}+24\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {4389\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {4389\,c}{128}+\frac {4389\,d\,x}{128}-\frac {368}{3}\right )\right )+\frac {19\,a^3\,\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 )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.84, size = 597, normalized size = 3.26 \[ \begin {cases} \frac {9 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {45 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {5 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {21 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {5 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {55 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} - \frac {a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {73 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \cos ^{11}{\left (c + d x \right )}}{693 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{21 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \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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