Optimal. Leaf size=109 \[ -\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5 a x}{128} \]
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Rubi [A] time = 0.12, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2838, 2565, 30, 2568, 2635, 8} \[ -\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5 a x}{128} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2565
Rule 2568
Rule 2635
Rule 2838
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin (c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} a \int \cos ^6(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} (5 a) \int \cos ^4(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} (5 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} (5 a) \int 1 \, dx\\ &=\frac {5 a x}{128}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 91, normalized size = 0.83 \[ -\frac {a (-336 \sin (2 (c+d x))+168 \sin (4 (c+d x))+112 \sin (6 (c+d x))+21 \sin (8 (c+d x))+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-840 d x)}{21504 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 73, normalized size = 0.67 \[ -\frac {384 \, a \cos \left (d x + c\right )^{7} - 105 \, a d x + 7 \, {\left (48 \, a \cos \left (d x + c\right )^{7} - 8 \, a \cos \left (d x + c\right )^{5} - 10 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 122, normalized size = 1.12 \[ \frac {5}{128} \, a x - \frac {a \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {3 \, a \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {5 \, a \cos \left (d x + c\right )}{64 \, d} - \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 78, normalized size = 0.72 \[ \frac {a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 63, normalized size = 0.58 \[ -\frac {3072 \, a \cos \left (d x + c\right )^{7} - 7 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{21504 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.39, size = 96, normalized size = 0.88 \[ -\frac {a\,\left (210\,\cos \left (c+d\,x\right )+126\,\cos \left (3\,c+3\,d\,x\right )+42\,\cos \left (5\,c+5\,d\,x\right )+6\,\cos \left (7\,c+7\,d\,x\right )-42\,\sin \left (2\,c+2\,d\,x\right )+21\,\sin \left (4\,c+4\,d\,x\right )+14\,\sin \left (6\,c+6\,d\,x\right )+\frac {21\,\sin \left (8\,c+8\,d\,x\right )}{8}-105\,d\,x\right )}{2688\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.23, size = 223, normalized size = 2.05 \[ \begin {cases} \frac {5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin {\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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