Optimal. Leaf size=31 \[ \frac{4 \cos ^5(a+b x)}{5 b}-\frac{4 \cos ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0496542, antiderivative size = 31, normalized size of antiderivative = 1., 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 = {4288, 2565, 14} \[ \frac{4 \cos ^5(a+b x)}{5 b}-\frac{4 \cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \sin (a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^2(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{4 \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{4 \cos ^3(a+b x)}{3 b}+\frac{4 \cos ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0662624, size = 27, normalized size = 0.87 \[ \frac{2 \cos ^3(a+b x) (3 \cos (2 (a+b x))-7)}{15 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 41, normalized size = 1.3 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{2\,b}}-{\frac{\cos \left ( 3\,bx+3\,a \right ) }{12\,b}}+{\frac{\cos \left ( 5\,bx+5\,a \right ) }{20\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23104, size = 49, normalized size = 1.58 \begin{align*} \frac{3 \, \cos \left (5 \, b x + 5 \, a\right ) - 5 \, \cos \left (3 \, b x + 3 \, a\right ) - 30 \, \cos \left (b x + a\right )}{60 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.478896, size = 62, normalized size = 2. \begin{align*} \frac{4 \,{\left (3 \, \cos \left (b x + a\right )^{5} - 5 \, \cos \left (b x + a\right )^{3}\right )}}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.35895, size = 92, normalized size = 2.97 \begin{align*} \begin{cases} - \frac{4 \sin{\left (a + b x \right )} \sin{\left (2 a + 2 b x \right )} \cos{\left (2 a + 2 b x \right )}}{15 b} - \frac{7 \sin ^{2}{\left (2 a + 2 b x \right )} \cos{\left (a + b x \right )}}{15 b} - \frac{8 \cos{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{15 b} & \text{for}\: b \neq 0 \\x \sin{\left (a \right )} \sin ^{2}{\left (2 a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29399, size = 54, normalized size = 1.74 \begin{align*} \frac{\cos \left (5 \, b x + 5 \, a\right )}{20 \, b} - \frac{\cos \left (3 \, b x + 3 \, a\right )}{12 \, b} - \frac{\cos \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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