Optimal. Leaf size=154 \[ -\frac {11 a^3 \cos ^7(c+d x)}{56 d}-\frac {11 \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{72 d}+\frac {11 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {55 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {55 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {55 a^3 x}{128}-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.15, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2678, 2669, 2635, 8} \[ -\frac {11 a^3 \cos ^7(c+d x)}{56 d}-\frac {11 \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{72 d}+\frac {11 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {55 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {55 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {55 a^3 x}{128}-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {1}{9} (11 a) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{72 d}+\frac {1}{8} \left (11 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {11 a^3 \cos ^7(c+d x)}{56 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{72 d}+\frac {1}{8} \left (11 a^3\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac {11 a^3 \cos ^7(c+d x)}{56 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{72 d}+\frac {1}{48} \left (55 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {11 a^3 \cos ^7(c+d x)}{56 d}+\frac {55 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{72 d}+\frac {1}{64} \left (55 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {11 a^3 \cos ^7(c+d x)}{56 d}+\frac {55 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {55 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{72 d}+\frac {1}{128} \left (55 a^3\right ) \int 1 \, dx\\ &=\frac {55 a^3 x}{128}-\frac {11 a^3 \cos ^7(c+d x)}{56 d}+\frac {55 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {55 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{72 d}\\ \end {align*}
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Mathematica [A] time = 2.00, size = 181, normalized size = 1.18 \[ -\frac {a^3 \left (6930 \sqrt {1-\sin (c+d x)} \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )+\sqrt {\sin (c+d x)+1} \left (896 \sin ^9(c+d x)+2128 \sin ^8(c+d x)-2000 \sin ^7(c+d x)-8248 \sin ^6(c+d x)-1224 \sin ^5(c+d x)+11514 \sin ^4(c+d x)+7174 \sin ^3(c+d x)-5641 \sin ^2(c+d x)-8311 \sin (c+d x)+3712\right )\right ) \cos ^7(c+d x)}{8064 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 98, normalized size = 0.64 \[ \frac {896 \, a^{3} \cos \left (d x + c\right )^{9} - 4608 \, a^{3} \cos \left (d x + c\right )^{7} + 3465 \, a^{3} d x - 21 \, {\left (144 \, a^{3} \cos \left (d x + c\right )^{7} - 88 \, a^{3} \cos \left (d x + c\right )^{5} - 110 \, a^{3} \cos \left (d x + c\right )^{3} - 165 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 157, normalized size = 1.02 \[ \frac {55}{128} \, a^{3} x + \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {9 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {29 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {33 \, a^{3} \cos \left (d x + c\right )}{128 \, d} - \frac {3 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {3 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 163, normalized size = 1.06 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 141, normalized size = 0.92 \[ -\frac {27648 \, a^{3} \cos \left (d x + c\right )^{7} - 1024 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 63 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 336 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{64512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.81, size = 501, normalized size = 3.25 \[ \frac {55\,a^3\,x}{128}-\frac {\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}-\frac {949\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}-\frac {699\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {699\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {949\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\frac {73\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{8064}-\frac {a^3\,\left (3465\,c+3465\,d\,x-7424\right )}{8064}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{896}-\frac {a^3\,\left (31185\,c+31185\,d\,x-18432\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{896}-\frac {a^3\,\left (31185\,c+31185\,d\,x-48384\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{224}-\frac {a^3\,\left (124740\,c+124740\,d\,x-129024\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{224}-\frac {a^3\,\left (124740\,c+124740\,d\,x-138240\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{96}-\frac {a^3\,\left (291060\,c+291060\,d\,x-236544\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{96}-\frac {a^3\,\left (291060\,c+291060\,d\,x-387072\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{64}-\frac {a^3\,\left (436590\,c+436590\,d\,x-290304\right )}{8064}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^3\,\left (3465\,c+3465\,d\,x\right )}{64}-\frac {a^3\,\left (436590\,c+436590\,d\,x-645120\right )}{8064}\right )-\frac {73\,a^3\,\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 )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.33, size = 439, normalized size = 2.85 \[ \begin {cases} \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {5 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {5 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {73 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} + \frac {5 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {15 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {11 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {3 a^{3} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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