Optimal. Leaf size=203 \[ \frac {a^3 \cos ^{11}(c+d x)}{11 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {5 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {5 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {15 a^3 x}{256} \]
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Rubi [A] time = 0.39, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 19, 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} \[ \frac {a^3 \cos ^{11}(c+d x)}{11 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {5 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {5 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {15 a^3 x}{256} \]
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))^3 \, dx &=\int \left (a^3 \cos ^4(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^5(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^6(c+d x)+a^3 \cos ^4(c+d x) \sin ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+a^3 \int \cos ^4(c+d x) \sin ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx\\ &=-\frac {a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{8} \left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{2} \left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int x^4 \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^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{16} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (9 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (x^4-3 x^6+3 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^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{64} \left (3 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 ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^3 x}{128}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{256} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac {15 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 1.04, size = 126, normalized size = 0.62 \[ \frac {a^3 (-13860 \sin (2 (c+d x))-46200 \sin (4 (c+d x))+6930 \sin (6 (c+d x))+5775 \sin (8 (c+d x))-1386 \sin (10 (c+d x))-198660 \cos (c+d x)-41580 \cos (3 (c+d x))+27258 \cos (5 (c+d x))+3630 \cos (7 (c+d x))-3850 \cos (9 (c+d x))+210 \cos (11 (c+d x))+138600 c+138600 d x)}{2365440 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 137, normalized size = 0.67 \[ \frac {26880 \, a^{3} \cos \left (d x + c\right )^{11} - 197120 \, a^{3} \cos \left (d x + c\right )^{9} + 380160 \, a^{3} \cos \left (d x + c\right )^{7} - 236544 \, a^{3} \cos \left (d x + c\right )^{5} + 17325 \, a^{3} d x - 231 \, {\left (384 \, a^{3} \cos \left (d x + c\right )^{9} - 1168 \, a^{3} \cos \left (d x + c\right )^{7} + 984 \, a^{3} \cos \left (d x + c\right )^{5} - 50 \, a^{3} \cos \left (d x + c\right )^{3} - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{295680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 191, normalized size = 0.94 \[ \frac {15}{256} \, a^{3} x + \frac {a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} - \frac {5 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{3072 \, d} + \frac {11 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {59 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac {9 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{512 \, d} - \frac {43 \, a^{3} \cos \left (d x + c\right )}{512 \, d} - \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {3 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 288, normalized size = 1.42 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 169, normalized size = 0.83 \[ \frac {2048 \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 22528 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 2310 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{2365440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.72, size = 506, normalized size = 2.49 \[ \frac {15\,a^3\,x}{256}-\frac {\frac {15\,a^3\,\left (c+d\,x\right )}{256}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}-\frac {242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {3987\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3987\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}-\frac {a^3\,\left (17325\,c+17325\,d\,x-53248\right )}{295680}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {165\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (190575\,c+190575\,d\,x-585728\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {825\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (952875\,c+952875\,d\,x-2928640\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (2858625\,c+2858625\,d\,x+675840\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (5717250\,c+5717250\,d\,x-3379200\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (2858625\,c+2858625\,d\,x-9461760\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (5717250\,c+5717250\,d\,x-14192640\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {3465\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (8004150\,c+8004150\,d\,x+16084992\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {3465\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (8004150\,c+8004150\,d\,x-40685568\right )}{295680}\right )+\frac {15\,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 = 50.88, size = 648, normalized size = 3.19 \[ \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 {3 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 {3 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 {9 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 {3 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 {3 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 {3 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {11 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {6 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {11 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {12 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {16 a^{3} \cos ^{11}{\left (c + d x \right )}}{1155 d} - \frac {8 a^{3} \cos ^{9}{\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \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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