Optimal. Leaf size=46 \[ \frac {16 \sin ^9(a+b x)}{9 b}-\frac {32 \sin ^7(a+b x)}{7 b}+\frac {16 \sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4287, 2564, 270} \[ \frac {16 \sin ^9(a+b x)}{9 b}-\frac {32 \sin ^7(a+b x)}{7 b}+\frac {16 \sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rule 4287
Rubi steps
\begin {align*} \int \cos (a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^5(a+b x) \sin ^4(a+b x) \, dx\\ &=\frac {16 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {16 \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {16 \sin ^5(a+b x)}{5 b}-\frac {32 \sin ^7(a+b x)}{7 b}+\frac {16 \sin ^9(a+b x)}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 37, normalized size = 0.80 \[ \frac {2 \sin ^5(a+b x) (220 \cos (2 (a+b x))+35 \cos (4 (a+b x))+249)}{315 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 53, normalized size = 1.15 \[ \frac {16 \, {\left (35 \, \cos \left (b x + a\right )^{8} - 50 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.92, size = 68, normalized size = 1.48 \[ \frac {\sin \left (9 \, b x + 9 \, a\right )}{144 \, b} + \frac {\sin \left (7 \, b x + 7 \, a\right )}{112 \, b} - \frac {\sin \left (5 \, b x + 5 \, a\right )}{20 \, b} - \frac {\sin \left (3 \, b x + 3 \, a\right )}{12 \, b} + \frac {3 \, \sin \left (b x + a\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.22, size = 69, normalized size = 1.50 \[ \frac {3 \sin \left (b x +a \right )}{8 b}-\frac {\sin \left (3 b x +3 a \right )}{12 b}-\frac {\sin \left (5 b x +5 a \right )}{20 b}+\frac {\sin \left (7 b x +7 a \right )}{112 b}+\frac {\sin \left (9 b x +9 a \right )}{144 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 58, normalized size = 1.26 \[ \frac {35 \, \sin \left (9 \, b x + 9 \, a\right ) + 45 \, \sin \left (7 \, b x + 7 \, a\right ) - 252 \, \sin \left (5 \, b x + 5 \, a\right ) - 420 \, \sin \left (3 \, b x + 3 \, a\right ) + 1890 \, \sin \left (b x + a\right )}{5040 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 36, normalized size = 0.78 \[ \frac {16\,\left (35\,{\sin \left (a+b\,x\right )}^9-90\,{\sin \left (a+b\,x\right )}^7+63\,{\sin \left (a+b\,x\right )}^5\right )}{315\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.31, size = 162, normalized size = 3.52 \[ \begin {cases} \frac {107 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )}}{315 b} + \frac {16 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{21 b} + \frac {128 \sin {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{315 b} - \frac {104 \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} - \frac {64 \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{315 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (2 a \right )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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