Optimal. Leaf size=153 \[ -\frac {a^2 \cos ^7(c+d x)}{28 d}-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{36 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac {5 a^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac {5 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {5 a^2 x}{64}-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.15, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac {a^2 \cos ^7(c+d x)}{28 d}-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{36 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac {5 a^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac {5 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {5 a^2 x}{64}-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rule 2860
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx &=-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {2}{9} \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac {1}{4} a \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x)}{28 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac {1}{4} a^2 \int \cos ^6(c+d x) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x)}{28 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac {1}{24} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x)}{28 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac {1}{32} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a^2 \cos ^7(c+d x)}{28 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}+\frac {1}{64} \left (5 a^2\right ) \int 1 \, dx\\ &=\frac {5 a^2 x}{64}-\frac {a^2 \cos ^7(c+d x)}{28 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{36 d}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 106, normalized size = 0.69 \[ \frac {a^2 (1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x))-3276 \cos (c+d x)-1848 \cos (3 (c+d x))-504 \cos (5 (c+d x))-18 \cos (7 (c+d x))+14 \cos (9 (c+d x))+2520 c+2520 d x)}{32256 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 98, normalized size = 0.64 \[ \frac {448 \, a^{2} \cos \left (d x + c\right )^{9} - 1152 \, a^{2} \cos \left (d x + c\right )^{7} + 315 \, a^{2} d x - 21 \, {\left (48 \, a^{2} \cos \left (d x + c\right )^{7} - 8 \, a^{2} \cos \left (d x + c\right )^{5} - 10 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4032 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 157, normalized size = 1.03 \[ \frac {5}{64} \, a^{2} x + \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {11 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {13 \, a^{2} \cos \left (d x + c\right )}{128 \, d} - \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 116, normalized size = 0.76 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+2 a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 93, normalized size = 0.61 \[ -\frac {4608 \, a^{2} \cos \left (d x + c\right )^{7} - 512 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{32256 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.85, size = 501, normalized size = 3.27 \[ \frac {5\,a^2\,x}{64}-\frac {\frac {83\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}-\frac {191\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{48}-\frac {145\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {145\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}-\frac {83\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}+\frac {191\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{48}-\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}+\frac {a^2\,\left (315\,c+315\,d\,x\right )}{4032}-\frac {a^2\,\left (315\,c+315\,d\,x-1408\right )}{4032}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{448}-\frac {a^2\,\left (2835\,c+2835\,d\,x-4608\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{448}-\frac {a^2\,\left (2835\,c+2835\,d\,x-8064\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{112}-\frac {a^2\,\left (11340\,c+11340\,d\,x-18432\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{112}-\frac {a^2\,\left (11340\,c+11340\,d\,x-32256\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{48}-\frac {a^2\,\left (26460\,c+26460\,d\,x-21504\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^2\,\left (39690\,c+39690\,d\,x-16128\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{48}-\frac {a^2\,\left (26460\,c+26460\,d\,x-96768\right )}{4032}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^2\,\left (39690\,c+39690\,d\,x-161280\right )}{4032}\right )+\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.87, size = 282, normalized size = 1.84 \[ \begin {cases} \frac {5 a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {5 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {5 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {55 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac {73 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin {\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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