Optimal. Leaf size=182 \[ -\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {21 a^3 x}{256} \]
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Rubi [A] time = 0.38, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 19, 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} \[ -\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {21 a^3 x}{256} \]
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 ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos ^4(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^5(c+d x)+a^3 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+a^3 \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {3 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{2} a^3 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+\frac {1}{8} \left (9 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{16} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{32} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{64} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{128} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac {9 a^3 x}{128}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {21 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{256} \left (3 a^3\right ) \int 1 \, dx\\ &=\frac {21 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {21 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 116, normalized size = 0.64 \[ \frac {a^3 (-60 \sin (2 (c+d x))-840 \sin (4 (c+d x))+30 \sin (6 (c+d x))+105 \sin (8 (c+d x))-6 \sin (10 (c+d x))-3600 \cos (c+d x)-960 \cos (3 (c+d x))+384 \cos (5 (c+d x))+120 \cos (7 (c+d x))-40 \cos (9 (c+d x))+2700 c+2520 d x)}{30720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 124, normalized size = 0.68 \[ -\frac {1280 \, a^{3} \cos \left (d x + c\right )^{9} - 3840 \, a^{3} \cos \left (d x + c\right )^{7} + 3072 \, a^{3} \cos \left (d x + c\right )^{5} - 315 \, a^{3} d x + 3 \, {\left (128 \, a^{3} \cos \left (d x + c\right )^{9} - 816 \, a^{3} \cos \left (d x + c\right )^{7} + 968 \, a^{3} \cos \left (d x + c\right )^{5} - 70 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 174, normalized size = 0.96 \[ \frac {21}{256} \, a^{3} x - \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{768 \, d} + \frac {a^{3} \cos \left (7 \, d x + 7 \, c\right )}{256 \, d} + \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a^{3} \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {15 \, a^{3} \cos \left (d x + c\right )}{128 \, d} - \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {7 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 252, normalized size = 1.38 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 149, normalized size = 0.82 \[ -\frac {2048 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 6144 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} + 21 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 630 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{215040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.81, size = 572, normalized size = 3.14 \[ \frac {21\,a^3\,x}{256}-\frac {\frac {203\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {1973\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {463\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {3231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {463\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {1973\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {203\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {a^3\,\left (315\,c+315\,d\,x\right )}{3840}-\frac {a^3\,\left (315\,c+315\,d\,x-1024\right )}{3840}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{384}-\frac {a^3\,\left (3150\,c+3150\,d\,x\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{384}-\frac {a^3\,\left (3150\,c+3150\,d\,x-10240\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{256}-\frac {a^3\,\left (14175\,c+14175\,d\,x-15360\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{256}-\frac {a^3\,\left (14175\,c+14175\,d\,x-30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^3\,\left (37800\,c+37800\,d\,x+30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {7\,a^3\,\left (315\,c+315\,d\,x\right )}{128}-\frac {a^3\,\left (66150\,c+66150\,d\,x+30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^3\,\left (37800\,c+37800\,d\,x-153600\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {21\,a^3\,\left (315\,c+315\,d\,x\right )}{320}-\frac {a^3\,\left (79380\,c+79380\,d\,x-129024\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {7\,a^3\,\left (315\,c+315\,d\,x\right )}{128}-\frac {a^3\,\left (66150\,c+66150\,d\,x-245760\right )}{3840}\right )+\frac {21\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 33.51, size = 595, normalized size = 3.27 \[ \begin {cases} \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {9 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {27 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {9 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {33 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {33 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {12 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {2 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \sin ^{3}{\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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