Optimal. Leaf size=183 \[ -\frac {a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^9(c+d x)}{3 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac {3 a^2 \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a^2 x}{128} \]
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Rubi [A] time = 0.27, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac {a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^9(c+d x)}{3 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac {3 a^2 \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a^2 x}{128} \]
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 ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^6(c+d x) \sin ^3(c+d x)+2 a^2 \cos ^6(c+d x) \sin ^4(c+d x)+a^2 \cos ^6(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{5} \left (3 a^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a^2 \operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^2 \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{40} \left (3 a^2\right ) \int \cos ^6(c+d x) \, dx-\frac {a^2 \operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos ^9(c+d x)}{3 d}-\frac {a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^2 \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{16} a^2 \int \cos ^4(c+d x) \, dx\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos ^9(c+d x)}{3 d}-\frac {a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^2 \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{64} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos ^9(c+d x)}{3 d}-\frac {a^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^2 \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{128} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac {3 a^2 x}{128}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos ^9(c+d x)}{3 d}-\frac {a^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^2 \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.99, size = 126, normalized size = 0.69 \[ \frac {a^2 (4620 \sin (2 (c+d x))-9240 \sin (4 (c+d x))-2310 \sin (6 (c+d x))+1155 \sin (8 (c+d x))+462 \sin (10 (c+d x))-39270 \cos (c+d x)-16170 \cos (3 (c+d x))+1155 \cos (5 (c+d x))+2805 \cos (7 (c+d x))+385 \cos (9 (c+d x))-105 \cos (11 (c+d x))+27720 c+27720 d x)}{1182720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 124, normalized size = 0.68 \[ -\frac {13440 \, a^{2} \cos \left (d x + c\right )^{11} - 49280 \, a^{2} \cos \left (d x + c\right )^{9} + 42240 \, a^{2} \cos \left (d x + c\right )^{7} - 3465 \, a^{2} d x - 231 \, {\left (128 \, a^{2} \cos \left (d x + c\right )^{9} - 176 \, a^{2} \cos \left (d x + c\right )^{7} + 8 \, a^{2} \cos \left (d x + c\right )^{5} + 10 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{147840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 191, normalized size = 1.04 \[ \frac {3}{128} \, a^{2} x - \frac {a^{2} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{3072 \, d} + \frac {17 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {7 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{512 \, d} - \frac {17 \, a^{2} \cos \left (d x + c\right )}{512 \, d} + \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{2560 \, d} + \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{512 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{256 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 172, normalized size = 0.94 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 116, normalized size = 0.63 \[ -\frac {5120 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 56320 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{3548160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.05, size = 543, normalized size = 2.97 \[ \frac {3\,a^2\,x}{128}-\frac {\frac {3\,a^2\,\left (c+d\,x\right )}{128}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {3323\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {108\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {841\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}+\frac {841\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {108\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {3323\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}-a^2\,\left (\frac {3\,c}{128}+\frac {3\,d\,x}{128}-\frac {20}{231}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {33\,a^2\,\left (c+d\,x\right )}{128}-a^2\,\left (\frac {33\,c}{128}+\frac {33\,d\,x}{128}-\frac {20}{21}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {165\,a^2\,\left (c+d\,x\right )}{128}-a^2\,\left (\frac {165\,c}{128}+\frac {165\,d\,x}{128}-4\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {165\,a^2\,\left (c+d\,x\right )}{128}-a^2\,\left (\frac {165\,c}{128}+\frac {165\,d\,x}{128}-\frac {16}{21}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {495\,a^2\,\left (c+d\,x\right )}{64}-a^2\,\left (\frac {495\,c}{64}+\frac {495\,d\,x}{64}+16\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {495\,a^2\,\left (c+d\,x\right )}{128}-a^2\,\left (\frac {495\,c}{128}+\frac {495\,d\,x}{128}-12\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {495\,a^2\,\left (c+d\,x\right )}{128}-a^2\,\left (\frac {495\,c}{128}+\frac {495\,d\,x}{128}-\frac {16}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {495\,a^2\,\left (c+d\,x\right )}{64}-a^2\,\left (\frac {495\,c}{64}+\frac {495\,d\,x}{64}-\frac {312}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {693\,a^2\,\left (c+d\,x\right )}{64}-a^2\,\left (\frac {693\,c}{64}+\frac {693\,d\,x}{64}+40\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {693\,a^2\,\left (c+d\,x\right )}{64}-a^2\,\left (\frac {693\,c}{64}+\frac {693\,d\,x}{64}-80\right )\right )+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{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 = 40.49, size = 384, normalized size = 2.10 \[ \begin {cases} \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{64} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{128} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{128 d} - \frac {8 a^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} - \frac {2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin ^{3}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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