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 = 0.0635506, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 4288
Rule 2565
Rule 270
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*}
Mathematica [A] time = 0.109167, size = 37, normalized size = 0.8 \[ \frac{\cos ^3(a+b x) (108 \cos (2 (a+b x))-15 \cos (4 (a+b x))-157)}{210 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 55, normalized size = 1.2 \begin{align*} -{\frac{5\,\cos \left ( bx+a \right ) }{16\,b}}-{\frac{\cos \left ( 3\,bx+3\,a \right ) }{48\,b}}+{\frac{3\,\cos \left ( 5\,bx+5\,a \right ) }{80\,b}}-{\frac{\cos \left ( 7\,bx+7\,a \right ) }{112\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14872, size = 63, normalized size = 1.37 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.485046, size = 95, normalized size = 2.07 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 73.1723, size = 202, normalized size = 4.39 \begin{align*} \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}{\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 = 1.30693, size = 73, normalized size = 1.59 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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