Optimal. Leaf size=224 \[ \frac {a^3 \cos ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac {9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac {27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac {9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac {27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {27 a^3 x}{1024} \]
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Rubi [A] time = 0.42, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 21, 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} \[ \frac {a^3 \cos ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac {9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac {27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac {9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac {27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {27 a^3 x}{1024} \]
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 ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos ^6(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^5(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^6(c+d x)+a^3 \cos ^6(c+d x) \sin ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx\\ &=-\frac {a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac {1}{10} \left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{4} \left (5 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^3 \, 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 )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac {1}{80} \left (3 a^3\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac {1}{32} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{64} \left (3 a^3\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac {1}{128} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{128} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac {9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac {1}{256} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{512} \left (15 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^3 x}{256}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac {9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac {\left (15 a^3\right ) \int 1 \, dx}{1024}\\ &=\frac {27 a^3 x}{1024}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac {9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 2.30, size = 146, normalized size = 0.65 \[ \frac {a^3 (80080 \sin (2 (c+d x))-385385 \sin (4 (c+d x))-40040 \sin (6 (c+d x))+65065 \sin (8 (c+d x))+8008 \sin (10 (c+d x))-5005 \sin (12 (c+d x))-1401400 \cos (c+d x)-450450 \cos (3 (c+d x))+150150 \cos (5 (c+d x))+94380 \cos (7 (c+d x))-20020 \cos (9 (c+d x))-11830 \cos (11 (c+d x))+770 \cos (13 (c+d x))+720720 c+1081080 d x)}{41000960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 150, normalized size = 0.67 \[ \frac {394240 \, a^{3} \cos \left (d x + c\right )^{13} - 2795520 \, a^{3} \cos \left (d x + c\right )^{11} + 5125120 \, a^{3} \cos \left (d x + c\right )^{9} - 2928640 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, a^{3} d x - 1001 \, {\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 3712 \, a^{3} \cos \left (d x + c\right )^{9} + 2864 \, a^{3} \cos \left (d x + c\right )^{7} - 72 \, a^{3} \cos \left (d x + c\right )^{5} - 90 \, a^{3} \cos \left (d x + c\right )^{3} - 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{5125120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 225, normalized size = 1.00 \[ \frac {27}{1024} \, a^{3} x + \frac {a^{3} \cos \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {13 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{45056 \, d} - \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{2048 \, d} + \frac {33 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{14336 \, d} + \frac {15 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {45 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{4096 \, d} - \frac {35 \, a^{3} \cos \left (d x + c\right )}{1024 \, d} - \frac {a^{3} \sin \left (12 \, d x + 12 \, c\right )}{8192 \, d} + \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {13 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {77 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 308, normalized size = 1.38 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{13}-\frac {6 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{143}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{429}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 184, normalized size = 0.82 \[ \frac {40960 \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 532480 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} + 12012 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 15015 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{123002880 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.19, size = 612, normalized size = 2.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.94, size = 748, normalized size = 3.34 \[ \begin {cases} \frac {15 a^{3} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {45 a^{3} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {225 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {75 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {225 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 a^{3} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{1024 d} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {99 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {99 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {2 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {85 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{1024 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{231 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {15 a^{3} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {16 a^{3} \cos ^{13}{\left (c + d x \right )}}{3003 d} - \frac {8 a^{3} \cos ^{11}{\left (c + d x \right )}}{231 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \sin ^{4}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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