Optimal. Leaf size=134 \[ -\frac {B (a \sin (c+d x)+a)^{10}}{10 a^8 d}-\frac {(A-7 B) (a \sin (c+d x)+a)^9}{9 a^7 d}+\frac {3 (A-3 B) (a \sin (c+d x)+a)^8}{4 a^6 d}-\frac {4 (3 A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}+\frac {4 (A-B) (a \sin (c+d x)+a)^6}{3 a^4 d} \]
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Rubi [A] time = 0.18, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 77} \[ -\frac {(A-7 B) (a \sin (c+d x)+a)^9}{9 a^7 d}+\frac {3 (A-3 B) (a \sin (c+d x)+a)^8}{4 a^6 d}-\frac {4 (3 A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}+\frac {4 (A-B) (a \sin (c+d x)+a)^6}{3 a^4 d}-\frac {B (a \sin (c+d x)+a)^{10}}{10 a^8 d} \]
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))^2 (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^5 \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)^5-4 a^2 (3 A-5 B) (a+x)^6+6 a (A-3 B) (a+x)^7+(-A+7 B) (a+x)^8-\frac {B (a+x)^9}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {4 (A-B) (a+a \sin (c+d x))^6}{3 a^4 d}-\frac {4 (3 A-5 B) (a+a \sin (c+d x))^7}{7 a^5 d}+\frac {3 (A-3 B) (a+a \sin (c+d x))^8}{4 a^6 d}-\frac {(A-7 B) (a+a \sin (c+d x))^9}{9 a^7 d}-\frac {B (a+a \sin (c+d x))^{10}}{10 a^8 d}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 86, normalized size = 0.64 \[ -\frac {a^2 (\sin (c+d x)+1)^6 \left (28 (5 A-17 B) \sin ^3(c+d x)+(651 B-525 A) \sin ^2(c+d x)+6 (115 A-61 B) \sin (c+d x)-325 A+126 B \sin ^4(c+d x)+61 B\right )}{1260 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 127, normalized size = 0.95 \[ \frac {126 \, B a^{2} \cos \left (d x + c\right )^{10} - 315 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{8} - 4 \, {\left (35 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{8} - 10 \, {\left (5 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{6} - 12 \, {\left (5 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{4} - 16 \, {\left (5 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} - 32 \, {\left (5 \, A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 239, normalized size = 1.78 \[ \frac {B a^{2} \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {A a^{2} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} + \frac {7 \, A a^{2} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (16 \, A a^{2} + 7 \, B a^{2}\right )} \cos \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {{\left (7 \, A a^{2} + 4 \, B a^{2}\right )} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac {7 \, {\left (8 \, A a^{2} + 5 \, B a^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {{\left (A a^{2} + 10 \, B a^{2}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (5 \, A a^{2} - 4 \, B a^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (11 \, A a^{2} + 2 \, B a^{2}\right )} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 231, normalized size = 1.72 \[ \frac {a^{2} A \left (-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{10}-\frac {\left (\cos ^{8}\left (d x +c \right )\right )}{40}\right )-\frac {a^{2} A \left (\cos ^{8}\left (d x +c \right )\right )}{4}+2 B \,a^{2} \left (-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )+\frac {a^{2} A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}-\frac {B \,a^{2} \left (\cos ^{8}\left (d x +c \right )\right )}{8}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 168, normalized size = 1.25 \[ -\frac {126 \, B a^{2} \sin \left (d x + c\right )^{10} + 140 \, {\left (A + 2 \, B\right )} a^{2} \sin \left (d x + c\right )^{9} + 315 \, {\left (A - B\right )} a^{2} \sin \left (d x + c\right )^{8} - 360 \, {\left (A + 3 \, B\right )} a^{2} \sin \left (d x + c\right )^{7} - 1260 \, A a^{2} \sin \left (d x + c\right )^{6} + 1512 \, B a^{2} \sin \left (d x + c\right )^{5} + 630 \, {\left (3 \, A + B\right )} a^{2} \sin \left (d x + c\right )^{4} + 840 \, {\left (A - B\right )} a^{2} \sin \left (d x + c\right )^{3} - 630 \, {\left (2 \, A + B\right )} a^{2} \sin \left (d x + c\right )^{2} - 1260 \, A a^{2} \sin \left (d x + c\right )}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.20, size = 168, normalized size = 1.25 \[ -\frac {\frac {2\,a^2\,{\sin \left (c+d\,x\right )}^3\,\left (A-B\right )}{3}-\frac {a^2\,{\sin \left (c+d\,x\right )}^2\,\left (2\,A+B\right )}{2}-A\,a^2\,{\sin \left (c+d\,x\right )}^6+\frac {a^2\,{\sin \left (c+d\,x\right )}^4\,\left (3\,A+B\right )}{2}+\frac {a^2\,{\sin \left (c+d\,x\right )}^8\,\left (A-B\right )}{4}-\frac {2\,a^2\,{\sin \left (c+d\,x\right )}^7\,\left (A+3\,B\right )}{7}+\frac {a^2\,{\sin \left (c+d\,x\right )}^9\,\left (A+2\,B\right )}{9}+\frac {6\,B\,a^2\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {B\,a^2\,{\sin \left (c+d\,x\right )}^{10}}{10}-A\,a^2\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.90, size = 440, normalized size = 3.28 \[ \begin {cases} \frac {16 A a^{2} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 A a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 A a^{2} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {2 A a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 A a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {2 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A a^{2} \cos ^{8}{\left (c + d x \right )}}{4 d} + \frac {B a^{2} \sin ^{10}{\left (c + d x \right )}}{40 d} + \frac {32 B a^{2} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {B a^{2} \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8 d} + \frac {16 B a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {B a^{2} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {4 B a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{4 d} + \frac {2 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right )^{2} \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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