Optimal. Leaf size=65 \[ -\frac {a \cos ^6(c+d x)}{6 d}+\frac {b \sin ^7(c+d x)}{7 d}-\frac {2 b \sin ^5(c+d x)}{5 d}+\frac {b \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.10, antiderivative size = 65, 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} \[ -\frac {a \cos ^6(c+d x)}{6 d}+\frac {b \sin ^7(c+d x)}{7 d}-\frac {2 b \sin ^5(c+d x)}{5 d}+\frac {b \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2564
Rule 2565
Rule 2834
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^5(c+d x) \sin (c+d x) \, dx+b \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^5 \, dx,x,\cos (c+d x)\right )}{d}+\frac {b \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {b \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {b \sin ^3(c+d x)}{3 d}-\frac {2 b \sin ^5(c+d x)}{5 d}+\frac {b \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 86, normalized size = 1.32 \[ -\frac {525 a \cos (2 (c+d x))+210 a \cos (4 (c+d x))+35 a \cos (6 (c+d x))+350 a-525 b \sin (c+d x)+35 b \sin (3 (c+d x))+63 b \sin (5 (c+d x))+15 b \sin (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 62, normalized size = 0.95 \[ -\frac {35 \, a \cos \left (d x + c\right )^{6} + 2 \, {\left (15 \, b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} - 4 \, b \cos \left (d x + c\right )^{2} - 8 \, b\right )} \sin \left (d x + c\right )}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 103, normalized size = 1.58 \[ -\frac {a \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} - \frac {b \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {3 \, b \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {b \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, b \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 64, normalized size = 0.98 \[ \frac {-\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{6}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 72, normalized size = 1.11 \[ \frac {30 \, b \sin \left (d x + c\right )^{7} + 35 \, a \sin \left (d x + c\right )^{6} - 84 \, b \sin \left (d x + c\right )^{5} - 105 \, a \sin \left (d x + c\right )^{4} + 70 \, b \sin \left (d x + c\right )^{3} + 105 \, a \sin \left (d x + c\right )^{2}}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.59, size = 71, normalized size = 1.09 \[ \frac {\frac {b\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}-\frac {2\,b\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^4}{2}+\frac {b\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.07, size = 90, normalized size = 1.38 \[ \begin {cases} - \frac {a \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {8 b \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right ) \sin {\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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