Optimal. Leaf size=134 \[ -\frac {B (a \sin (c+d x)+a)^9}{9 a^8 d}-\frac {(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac {6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac {2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac {8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d} \]
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Rubi [A] time = 0.14, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2836, 77} \[ -\frac {(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac {6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac {2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac {8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d}-\frac {B (a \sin (c+d x)+a)^9}{9 a^8 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^4 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (A-B) (a+x)^4-4 a^2 (3 A-5 B) (a+x)^5+6 a (A-3 B) (a+x)^6+(-A+7 B) (a+x)^7-\frac {B (a+x)^8}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {8 (A-B) (a+a \sin (c+d x))^5}{5 a^4 d}-\frac {2 (3 A-5 B) (a+a \sin (c+d x))^6}{3 a^5 d}+\frac {6 (A-3 B) (a+a \sin (c+d x))^7}{7 a^6 d}-\frac {(A-7 B) (a+a \sin (c+d x))^8}{8 a^7 d}-\frac {B (a+a \sin (c+d x))^9}{9 a^8 d}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 194, normalized size = 1.45 \[ \frac {a (\sin (c+d x)+1) (-17640 (A+B) \cos (2 (c+d x))-8820 (A+B) \cos (4 (c+d x))+176400 A \sin (c+d x)+35280 A \sin (3 (c+d x))+7056 A \sin (5 (c+d x))+720 A \sin (7 (c+d x))-2520 A \cos (6 (c+d x))-315 A \cos (8 (c+d x))+17640 B \sin (c+d x)-2016 B \sin (5 (c+d x))-900 B \sin (7 (c+d x))-140 B \sin (9 (c+d x))-2520 B \cos (6 (c+d x))-315 B \cos (8 (c+d x)))}{322560 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 97, normalized size = 0.72 \[ -\frac {315 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, B a \cos \left (d x + c\right )^{8} - 5 \, {\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{6} - 6 \, {\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{4} - 8 \, {\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{2} - 16 \, {\left (9 \, A + B\right )} a\right )} \sin \left (d x + c\right )}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 182, normalized size = 1.36 \[ -\frac {B a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {7 \, A a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (A a + B a\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (A a + B a\right )} \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, {\left (A a + B a\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, {\left (A a + B a\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (4 \, A a - 5 \, B a\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A a - 2 \, B a\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (10 \, A a + B a\right )} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 128, normalized size = 0.96 \[ \frac {a B \left (-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {a A \left (\cos ^{8}\left (d x +c \right )\right )}{8}-\frac {a B \left (\cos ^{8}\left (d x +c \right )\right )}{8}+\frac {a A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 134, normalized size = 1.00 \[ -\frac {280 \, B a \sin \left (d x + c\right )^{9} + 315 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{8} + 360 \, {\left (A - 3 \, B\right )} a \sin \left (d x + c\right )^{7} - 1260 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{6} - 1512 \, {\left (A - B\right )} a \sin \left (d x + c\right )^{5} + 1890 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{4} + 840 \, {\left (3 \, A - B\right )} a \sin \left (d x + c\right )^{3} - 1260 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{2} - 2520 \, A a \sin \left (d x + c\right )}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 134, normalized size = 1.00 \[ -\frac {\frac {B\,a\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,\left (A-3\,B\right )\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^6}{2}-\frac {3\,a\,\left (A-B\right )\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,\left (3\,A-B\right )\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^2}{2}-A\,a\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.88, size = 228, normalized size = 1.70 \[ \begin {cases} \frac {16 A a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A a \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {16 B a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 B a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 B a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {B a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right ) \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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