Optimal. Leaf size=28 \[ \frac {\cos ^8(a+b x)}{b}-\frac {4 \cos ^6(a+b x)}{3 b} \]
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Rubi [A] time = 0.06, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4287, 2565, 14} \[ \frac {\cos ^8(a+b x)}{b}-\frac {4 \cos ^6(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2565
Rule 4287
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^5(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac {8 \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {8 \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {4 \cos ^6(a+b x)}{3 b}+\frac {\cos ^8(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 48, normalized size = 1.71 \[ \frac {-72 \cos (2 (a+b x))-12 \cos (4 (a+b x))+8 \cos (6 (a+b x))+3 \cos (8 (a+b x))}{384 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 26, normalized size = 0.93 \[ \frac {3 \, \cos \left (b x + a\right )^{8} - 4 \, \cos \left (b x + a\right )^{6}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 57, normalized size = 2.04 \[ \frac {\cos \left (8 \, b x + 8 \, a\right )}{128 \, b} + \frac {\cos \left (6 \, b x + 6 \, a\right )}{48 \, b} - \frac {\cos \left (4 \, b x + 4 \, a\right )}{32 \, b} - \frac {3 \, \cos \left (2 \, b x + 2 \, a\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 58, normalized size = 2.07 \[ -\frac {3 \cos \left (2 b x +2 a \right )}{16 b}-\frac {\cos \left (4 b x +4 a \right )}{32 b}+\frac {\cos \left (6 b x +6 a \right )}{48 b}+\frac {\cos \left (8 b x +8 a \right )}{128 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 50, normalized size = 1.79 \[ \frac {3 \, \cos \left (8 \, b x + 8 \, a\right ) + 8 \, \cos \left (6 \, b x + 6 \, a\right ) - 12 \, \cos \left (4 \, b x + 4 \, a\right ) - 72 \, \cos \left (2 \, b x + 2 \, a\right )}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 26, normalized size = 0.93 \[ -\frac {\frac {4\,{\cos \left (a+b\,x\right )}^6}{3}-{\cos \left (a+b\,x\right )}^8}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 39.28, size = 359, normalized size = 12.82 \[ \begin {cases} - \frac {3 x \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )}}{16} - \frac {3 x \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{16} - \frac {3 x \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{8} - \frac {3 x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{8} + \frac {3 x \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {3 x \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{16} - \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{96 b} - \frac {3 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{16 b} - \frac {\sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{8 b} - \frac {\sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{2 b} - \frac {31 \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{96 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (2 a \right )} \cos ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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