Optimal. Leaf size=46 \[ -\frac {4 \cos ^7(a+b x)}{7 b}+\frac {8 \cos ^5(a+b x)}{5 b}-\frac {4 \cos ^3(a+b x)}{3 b} \]
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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 = {4288, 2565, 270} \[ -\frac {4 \cos ^7(a+b x)}{7 b}+\frac {8 \cos ^5(a+b x)}{5 b}-\frac {4 \cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^2(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac {4 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {4 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {4 \cos ^3(a+b x)}{3 b}+\frac {8 \cos ^5(a+b x)}{5 b}-\frac {4 \cos ^7(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 37, normalized size = 0.80 \[ \frac {\cos ^3(a+b x) (108 \cos (2 (a+b x))-15 \cos (4 (a+b x))-157)}{210 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 36, normalized size = 0.78 \[ -\frac {4 \, {\left (15 \, \cos \left (b x + a\right )^{7} - 42 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3}\right )}}{105 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 54, normalized size = 1.17 \[ -\frac {\cos \left (7 \, b x + 7 \, a\right )}{112 \, b} + \frac {3 \, \cos \left (5 \, b x + 5 \, a\right )}{80 \, b} - \frac {\cos \left (3 \, b x + 3 \, a\right )}{48 \, b} - \frac {5 \, \cos \left (b x + a\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 55, normalized size = 1.20 \[ -\frac {5 \cos \left (b x +a \right )}{16 b}-\frac {\cos \left (3 b x +3 a \right )}{48 b}+\frac {3 \cos \left (5 b x +5 a \right )}{80 b}-\frac {\cos \left (7 b x +7 a \right )}{112 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 47, normalized size = 1.02 \[ -\frac {15 \, \cos \left (7 \, b x + 7 \, a\right ) - 63 \, \cos \left (5 \, b x + 5 \, a\right ) + 35 \, \cos \left (3 \, b x + 3 \, a\right ) + 525 \, \cos \left (b x + a\right )}{1680 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 36, normalized size = 0.78 \[ -\frac {4\,\left (15\,{\cos \left (a+b\,x\right )}^7-42\,{\cos \left (a+b\,x\right )}^5+35\,{\cos \left (a+b\,x\right )}^3\right )}{105\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.63, size = 202, normalized size = 4.39 \[ \begin {cases} - \frac {12 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{35 b} - \frac {11 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{35 b} - \frac {24 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} + \frac {8 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{35 b} - \frac {38 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{105 b} - \frac {32 \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{105 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\relax (a )} \sin ^{2}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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