Optimal. Leaf size=81 \[ \frac {a \sin ^7(c+d x)}{7 d}-\frac {2 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}+\frac {b \cos ^8(c+d x)}{8 d}-\frac {b \cos ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.14, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2564, 270, 2565, 14} \[ \frac {a \sin ^7(c+d x)}{7 d}-\frac {2 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}+\frac {b \cos ^8(c+d x)}{8 d}-\frac {b \cos ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2564
Rule 2565
Rule 2834
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx+b \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}-\frac {b \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {b \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {b \cos ^6(c+d x)}{6 d}+\frac {b \cos ^8(c+d x)}{8 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {2 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 94, normalized size = 1.16 \[ -\frac {-8400 a \sin (c+d x)+560 a \sin (3 (c+d x))+1008 a \sin (5 (c+d x))+240 a \sin (7 (c+d x))+2520 b \cos (2 (c+d x))+420 b \cos (4 (c+d x))-280 b \cos (6 (c+d x))-105 b \cos (8 (c+d x))}{107520 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 73, normalized size = 0.90 \[ \frac {105 \, b \cos \left (d x + c\right )^{8} - 140 \, b \cos \left (d x + c\right )^{6} - 8 \, {\left (15 \, a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} - 8 \, a\right )} \sin \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 118, normalized size = 1.46 \[ \frac {b \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {b \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {b \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, b \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {3 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, a \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 84, normalized size = 1.04 \[ \frac {a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 72, normalized size = 0.89 \[ \frac {105 \, b \sin \left (d x + c\right )^{8} + 120 \, a \sin \left (d x + c\right )^{7} - 280 \, b \sin \left (d x + c\right )^{6} - 336 \, a \sin \left (d x + c\right )^{5} + 210 \, b \sin \left (d x + c\right )^{4} + 280 \, a \sin \left (d x + c\right )^{3}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 71, normalized size = 0.88 \[ \frac {\frac {b\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {b\,{\sin \left (c+d\,x\right )}^6}{3}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {b\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.39, size = 136, normalized size = 1.68 \[ \begin {cases} \frac {8 a \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} + \frac {b \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {b \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {b \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right ) \sin ^{2}{\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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