Optimal. Leaf size=105 \[ \frac {B (a \sin (c+d x)+a)^8}{8 a^6 d}+\frac {(A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac {2 (A-2 B) (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac {4 (A-B) (a \sin (c+d x)+a)^5}{5 a^3 d} \]
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Rubi [A] time = 0.15, antiderivative size = 105, 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} \[ \frac {(A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac {2 (A-2 B) (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac {4 (A-B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac {B (a \sin (c+d x)+a)^8}{8 a^6 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^4 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (A-B) (a+x)^4-4 a (A-2 B) (a+x)^5+(A-5 B) (a+x)^6+\frac {B (a+x)^7}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {4 (A-B) (a+a \sin (c+d x))^5}{5 a^3 d}-\frac {2 (A-2 B) (a+a \sin (c+d x))^6}{3 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^7}{7 a^5 d}+\frac {B (a+a \sin (c+d x))^8}{8 a^6 d}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 70, normalized size = 0.67 \[ \frac {a^2 (\sin (c+d x)+1)^5 \left (15 (8 A-19 B) \sin ^2(c+d x)-5 (64 A-47 B) \sin (c+d x)+232 A+105 B \sin ^3(c+d x)-47 B\right )}{840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 109, normalized size = 1.04 \[ \frac {105 \, B a^{2} \cos \left (d x + c\right )^{8} - 280 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{6} - 8 \, {\left (15 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 6 \, {\left (4 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{4} - 8 \, {\left (4 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} - 16 \, {\left (4 \, A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 202, normalized size = 1.92 \[ \frac {B a^{2} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (4 \, A a^{2} + B a^{2}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (16 \, A a^{2} + 9 \, B a^{2}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (20 \, A a^{2} + 13 \, B a^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (A a^{2} - 6 \, B a^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (19 \, A a^{2} - 2 \, B a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (9 \, A a^{2} + 2 \, B a^{2}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 201, normalized size = 1.91 \[ \frac {a^{2} A \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )-\frac {a^{2} A \left (\cos ^{6}\left (d x +c \right )\right )}{3}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+\frac {a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}-\frac {B \,a^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{6}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 142, normalized size = 1.35 \[ \frac {105 \, B a^{2} \sin \left (d x + c\right )^{8} + 120 \, {\left (A + 2 \, B\right )} a^{2} \sin \left (d x + c\right )^{7} + 140 \, {\left (2 \, A - B\right )} a^{2} \sin \left (d x + c\right )^{6} - 168 \, {\left (A + 4 \, B\right )} a^{2} \sin \left (d x + c\right )^{5} - 210 \, {\left (4 \, A + B\right )} a^{2} \sin \left (d x + c\right )^{4} - 280 \, {\left (A - 2 \, B\right )} a^{2} \sin \left (d x + c\right )^{3} + 420 \, {\left (2 \, A + B\right )} a^{2} \sin \left (d x + c\right )^{2} + 840 \, A a^{2} \sin \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 140, normalized size = 1.33 \[ \frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^2\,\left (2\,A+B\right )}{2}-\frac {a^2\,{\sin \left (c+d\,x\right )}^3\,\left (A-2\,B\right )}{3}-\frac {a^2\,{\sin \left (c+d\,x\right )}^4\,\left (4\,A+B\right )}{4}-\frac {a^2\,{\sin \left (c+d\,x\right )}^5\,\left (A+4\,B\right )}{5}+\frac {a^2\,{\sin \left (c+d\,x\right )}^7\,\left (A+2\,B\right )}{7}+\frac {B\,a^2\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a^2\,{\sin \left (c+d\,x\right )}^6\,\left (2\,A-B\right )}{6}+A\,a^2\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.17, size = 335, normalized size = 3.19 \[ \begin {cases} \frac {8 A a^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 A a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {8 A a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} + \frac {4 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a^{2} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {B a^{2} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {16 B a^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {B a^{2} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {8 B a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {B a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {2 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right )^{2} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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