Optimal. Leaf size=143 \[ \frac {(a B+A b) \sin ^6(c+d x)}{6 d}+\frac {(a A-2 b B) \sin ^5(c+d x)}{5 d}-\frac {(a B+A b) \sin ^4(c+d x)}{2 d}-\frac {(2 a A-b B) \sin ^3(c+d x)}{3 d}+\frac {(a B+A b) \sin ^2(c+d x)}{2 d}+\frac {a A \sin (c+d x)}{d}+\frac {b B \sin ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.17, antiderivative size = 143, 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 = {2837, 772} \[ \frac {(a B+A b) \sin ^6(c+d x)}{6 d}+\frac {(a A-2 b B) \sin ^5(c+d x)}{5 d}-\frac {(a B+A b) \sin ^4(c+d x)}{2 d}-\frac {(2 a A-b B) \sin ^3(c+d x)}{3 d}+\frac {(a B+A b) \sin ^2(c+d x)}{2 d}+\frac {a A \sin (c+d x)}{d}+\frac {b B \sin ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a+x) \left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a A b^4+b^3 (A b+a B) x+b^2 (-2 a A+b B) x^2-2 b (A b+a B) x^3+(a A-2 b B) x^4+\frac {(A b+a B) x^5}{b}+\frac {B x^6}{b}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {a A \sin (c+d x)}{d}+\frac {(A b+a B) \sin ^2(c+d x)}{2 d}-\frac {(2 a A-b B) \sin ^3(c+d x)}{3 d}-\frac {(A b+a B) \sin ^4(c+d x)}{2 d}+\frac {(a A-2 b B) \sin ^5(c+d x)}{5 d}+\frac {(A b+a B) \sin ^6(c+d x)}{6 d}+\frac {b B \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 116, normalized size = 0.81 \[ \frac {\sin (c+d x) \left (35 (a B+A b) \sin ^5(c+d x)+42 (a A-2 b B) \sin ^4(c+d x)-105 (a B+A b) \sin ^3(c+d x)-70 (2 a A-b B) \sin ^2(c+d x)+105 (a B+A b) \sin (c+d x)+210 a A+30 b B \sin ^6(c+d x)\right )}{210 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 88, normalized size = 0.62 \[ -\frac {35 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (15 \, B b \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, A a + B b\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (7 \, A a + B b\right )} \cos \left (d x + c\right )^{2} - 56 \, A a - 8 \, B b\right )} \sin \left (d x + c\right )}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 145, normalized size = 1.01 \[ -\frac {B b \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (B a + A b\right )} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {{\left (B a + A b\right )} \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {5 \, {\left (B a + A b\right )} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (4 \, A a - 3 \, B b\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (20 \, A a - B b\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (8 \, A a + B b\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 108, normalized size = 0.76 \[ \frac {B b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {A b \left (\cos ^{6}\left (d x +c \right )\right )}{6}-\frac {a B \left (\cos ^{6}\left (d x +c \right )\right )}{6}+\frac {a A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 116, normalized size = 0.81 \[ \frac {30 \, B b \sin \left (d x + c\right )^{7} + 35 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{6} + 42 \, {\left (A a - 2 \, B b\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{4} - 70 \, {\left (2 \, A a - B b\right )} \sin \left (d x + c\right )^{3} + 210 \, A a \sin \left (d x + c\right ) + 105 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{2}}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.06, size = 118, normalized size = 0.83 \[ \frac {\frac {B\,b\,{\sin \left (c+d\,x\right )}^7}{7}+\left (\frac {A\,b}{6}+\frac {B\,a}{6}\right )\,{\sin \left (c+d\,x\right )}^6+\left (\frac {A\,a}{5}-\frac {2\,B\,b}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^4+\left (\frac {B\,b}{3}-\frac {2\,A\,a}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^2+A\,a\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.84, size = 178, normalized size = 1.24 \[ \begin {cases} \frac {8 A a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A b \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {B a \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {8 B b \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 B b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {B b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a + b \sin {\relax (c )}\right ) \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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