Optimal. Leaf size=81 \[ -\frac {a \sin ^9(c+d x)}{9 d}+\frac {3 a \sin ^7(c+d x)}{7 d}-\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2834, 2565, 30, 2564, 270} \[ -\frac {a \sin ^9(c+d x)}{9 d}+\frac {3 a \sin ^7(c+d x)}{7 d}-\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2564
Rule 2565
Rule 2834
Rubi steps
\begin {align*} \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^7(c+d x) \sin (c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \operatorname {Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 60, normalized size = 0.74 \[ \frac {a \left (\sin ^3(c+d x) (1389 \cos (2 (c+d x))+330 \cos (4 (c+d x))+35 \cos (6 (c+d x))+1606)-1260 \cos ^8(c+d x)\right )}{10080 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 73, normalized size = 0.90 \[ -\frac {315 \, a \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 118, normalized size = 1.46 \[ -\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {5 \, a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{160 \, d} + \frac {7 \, a \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 74, normalized size = 0.91 \[ \frac {a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{8}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 94, normalized size = 1.16 \[ -\frac {280 \, a \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 1080 \, a \sin \left (d x + c\right )^{7} - 1260 \, a \sin \left (d x + c\right )^{6} + 1512 \, a \sin \left (d x + c\right )^{5} + 1890 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 1260 \, a \sin \left (d x + c\right )^{2}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.02, size = 93, normalized size = 1.15 \[ \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.73, size = 114, normalized size = 1.41 \[ \begin {cases} \frac {16 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin {\relax (c )} \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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