Optimal. Leaf size=209 \[ -\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac {17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac {17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac {17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {17 a^2 x}{1024} \]
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Rubi [A] time = 0.40, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 18, 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} \[ -\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac {17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac {17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac {17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {17 a^2 x}{1024} \]
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))^2 \, dx &=\int \left (a^2 \cos ^6(c+d x) \sin ^4(c+d x)+2 a^2 \cos ^6(c+d x) \sin ^5(c+d x)+a^2 \cos ^6(c+d x) \sin ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{10} \left (3 a^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{12} \left (5 a^2\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx-\frac {\left (2 a^2\right ) \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)}{80 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{80} \left (3 a^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} a^2 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \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 {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{64} a^2 \int \cos ^6(c+d x) \, dx+\frac {1}{32} a^2 \int \cos ^4(c+d x) \, dx\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{384} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{128} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{512} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{256} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac {3 a^2 x}{256}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {\left (5 a^2\right ) \int 1 \, dx}{1024}\\ &=\frac {17 a^2 x}{1024}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}\\ \end {align*}
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Mathematica [A] time = 1.39, size = 136, normalized size = 0.65 \[ \frac {a^2 (55440 \sin (2 (c+d x))-162855 \sin (4 (c+d x))-27720 \sin (6 (c+d x))+24255 \sin (8 (c+d x))+5544 \sin (10 (c+d x))-1155 \sin (12 (c+d x))-554400 \cos (c+d x)-184800 \cos (3 (c+d x))+55440 \cos (5 (c+d x))+39600 \cos (7 (c+d x))-6160 \cos (9 (c+d x))-5040 \cos (11 (c+d x))+166320 c+471240 d x)}{28385280 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 137, normalized size = 0.66 \[ -\frac {645120 \, a^{2} \cos \left (d x + c\right )^{11} - 1576960 \, a^{2} \cos \left (d x + c\right )^{9} + 1013760 \, a^{2} \cos \left (d x + c\right )^{7} - 58905 \, a^{2} d x + 231 \, {\left (1280 \, a^{2} \cos \left (d x + c\right )^{11} - 4736 \, a^{2} \cos \left (d x + c\right )^{9} + 4272 \, a^{2} \cos \left (d x + c\right )^{7} - 136 \, a^{2} \cos \left (d x + c\right )^{5} - 170 \, a^{2} \cos \left (d x + c\right )^{3} - 255 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 208, normalized size = 1.00 \[ \frac {17}{1024} \, a^{2} x - \frac {a^{2} \cos \left (11 \, d x + 11 \, c\right )}{5632 \, d} - \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{4608 \, d} + \frac {5 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{3584 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{512 \, d} - \frac {5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{768 \, d} - \frac {5 \, a^{2} \cos \left (d x + c\right )}{256 \, d} - \frac {a^{2} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {7 \, a^{2} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {47 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 238, normalized size = 1.14 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 138, normalized size = 0.66 \[ -\frac {81920 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 2772 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} - 1155 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{28385280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.12, size = 518, normalized size = 2.48 \[ \frac {a^2\,\left (\frac {17\,c}{1024}-\frac {17\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}-\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{231}-\frac {595\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{21}+\frac {11097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2560}+\frac {704\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{63}+\frac {27449\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2560}-\frac {384\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {202307\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3840}+\frac {192\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{7}+\frac {28659\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{256}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {28659\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{256}-64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {202307\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{3840}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {27449\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{2560}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{3}-\frac {11097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2560}+\frac {595\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{1536}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{23}}{512}+\frac {17\,d\,x}{1024}+\frac {51\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (c+d\,x\right )}{256}+\frac {561\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (c+d\,x\right )}{512}+\frac {935\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (c+d\,x\right )}{256}+\frac {8415\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (c+d\,x\right )}{1024}+\frac {1683\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (c+d\,x\right )}{128}+\frac {3927\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (c+d\,x\right )}{256}+\frac {1683\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (c+d\,x\right )}{128}+\frac {8415\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (c+d\,x\right )}{1024}+\frac {935\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (c+d\,x\right )}{256}+\frac {561\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}\,\left (c+d\,x\right )}{512}+\frac {51\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}\,\left (c+d\,x\right )}{256}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{24}\,\left (c+d\,x\right )}{1024}-\frac {32}{693}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 63.18, size = 656, normalized size = 3.14 \[ \begin {cases} \frac {5 a^{2} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {15 a^{2} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {75 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {25 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {75 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {5 a^{2} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a^{2} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 a^{2} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {33 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {2 a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {85 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {16 a^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \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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