Optimal. Leaf size=46 \[ \frac {4 \sin ^7(a+b x)}{7 b}-\frac {8 \sin ^5(a+b x)}{5 b}+\frac {4 \sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.06, antiderivative size = 46, 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, 2564, 270} \[ \frac {4 \sin ^7(a+b x)}{7 b}-\frac {8 \sin ^5(a+b x)}{5 b}+\frac {4 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rule 4287
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^5(a+b x) \sin ^2(a+b x) \, dx\\ &=\frac {4 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {4 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b}+\frac {4 \sin ^7(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 37, normalized size = 0.80 \[ \frac {\sin ^3(a+b x) (108 \cos (2 (a+b x))+15 \cos (4 (a+b x))+157)}{210 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 43, normalized size = 0.93 \[ -\frac {4 \, {\left (15 \, \cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 8\right )} \sin \left (b x + a\right )}{105 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 54, normalized size = 1.17 \[ -\frac {\sin \left (7 \, b x + 7 \, a\right )}{112 \, b} - \frac {3 \, \sin \left (5 \, b x + 5 \, a\right )}{80 \, b} - \frac {\sin \left (3 \, b x + 3 \, a\right )}{48 \, b} + \frac {5 \, \sin \left (b x + a\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.99, size = 55, normalized size = 1.20 \[ \frac {5 \sin \left (b x +a \right )}{16 b}-\frac {\sin \left (3 b x +3 a \right )}{48 b}-\frac {3 \sin \left (5 b x +5 a \right )}{80 b}-\frac {\sin \left (7 b x +7 a \right )}{112 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 47, normalized size = 1.02 \[ -\frac {15 \, \sin \left (7 \, b x + 7 \, a\right ) + 63 \, \sin \left (5 \, b x + 5 \, a\right ) + 35 \, \sin \left (3 \, b x + 3 \, a\right ) - 525 \, \sin \left (b x + a\right )}{1680 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 36, normalized size = 0.78 \[ \frac {4\,\left (15\,{\sin \left (a+b\,x\right )}^7-42\,{\sin \left (a+b\,x\right )}^5+35\,{\sin \left (a+b\,x\right )}^3\right )}{105\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.32, size = 202, normalized size = 4.39 \[ \begin {cases} \frac {38 \sin ^{3}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )}}{105 b} + \frac {32 \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{105 b} + \frac {8 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{35 b} + \frac {11 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{35 b} + \frac {24 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} - \frac {12 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{35 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (2 a \right )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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