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 = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 14
Rule 2565
Rule 4288
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.51, size = 26, normalized size = 0.84 \[ \frac {4 \, {\left (3 \, \cos \left (b x + a\right )^{5} - 5 \, \cos \left (b x + a\right )^{3}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 40, normalized size = 1.29 \[ \frac {\cos \left (5 \, b x + 5 \, a\right )}{20 \, b} - \frac {\cos \left (3 \, b x + 3 \, a\right )}{12 \, b} - \frac {\cos \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 41, normalized size = 1.32 \[ -\frac {\cos \left (b x +a \right )}{2 b}-\frac {\cos \left (3 b x +3 a \right )}{12 b}+\frac {\cos \left (5 b x +5 a \right )}{20 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 36, normalized size = 1.16 \[ \frac {3 \, \cos \left (5 \, b x + 5 \, a\right ) - 5 \, \cos \left (3 \, b x + 3 \, a\right ) - 30 \, \cos \left (b x + a\right )}{60 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 26, normalized size = 0.84 \[ -\frac {4\,\left (5\,{\cos \left (a+b\,x\right )}^3-3\,{\cos \left (a+b\,x\right )}^5\right )}{15\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.40, size = 92, normalized size = 2.97 \[ \begin {cases} - \frac {4 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{15 b} - \frac {7 \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{15 b} - \frac {8 \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \sin {\relax (a )} \sin ^{2}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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