Optimal. Leaf size=165 \[ -\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]
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Rubi [A] time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2568, 2635, 8, 2565, 270} \[ -\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{10} (3 a) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{80} (3 a) \int \cos ^6(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{32} a \int \cos ^4(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{128} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{256} (3 a) \int 1 \, dx\\ &=\frac {3 a x}{256}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 121, normalized size = 0.73 \[ \frac {a (13860 \sin (2 (c+d x))-27720 \sin (4 (c+d x))-6930 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+1386 \sin (10 (c+d x))-69300 \cos (c+d x)-23100 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+4950 \cos (7 (c+d x))-770 \cos (9 (c+d x))-630 \cos (11 (c+d x))+83160 d x)}{7096320 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 106, normalized size = 0.64 \[ -\frac {80640 \, a \cos \left (d x + c\right )^{11} - 197120 \, a \cos \left (d x + c\right )^{9} + 126720 \, a \cos \left (d x + c\right )^{7} - 10395 \, a d x - 693 \, {\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 167, normalized size = 1.01 \[ \frac {3}{256} \, a x - \frac {a \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} - \frac {a \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {5 \, a \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {5 \, a \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {5 \, a \cos \left (d x + c\right )}{512 \, d} + \frac {a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 134, normalized size = 0.81 \[ \frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 86, normalized size = 0.52 \[ -\frac {10240 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{7096320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.87, size = 447, normalized size = 2.71 \[ \frac {3\,a\,x}{256}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {3323\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}+\left (\frac {a\,\left (1715175\,c+1715175\,d\,x-9461760\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\left (\frac {a\,\left (3430350\,c+3430350\,d\,x+23654400\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {841\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\left (\frac {a\,\left (4802490\,c+4802490\,d\,x-52039680\right )}{887040}-\frac {693\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {a\,\left (4802490\,c+4802490\,d\,x+42577920\right )}{887040}-\frac {693\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {841\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\left (\frac {a\,\left (3430350\,c+3430350\,d\,x-30412800\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\left (\frac {a\,\left (1715175\,c+1715175\,d\,x+6082560\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {3323\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}+\left (\frac {a\,\left (571725\,c+571725\,d\,x-1126400\right )}{887040}-\frac {165\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\left (\frac {a\,\left (114345\,c+114345\,d\,x-225280\right )}{887040}-\frac {33\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {a\,\left (10395\,c+10395\,d\,x-20480\right )}{887040}-\frac {3\,a\,\left (c+d\,x\right )}{256}}{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 = 44.94, size = 318, normalized size = 1.93 \[ \begin {cases} \frac {3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {4 a \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {8 a \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{4}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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