Optimal. Leaf size=31 \[ \frac {\sin ^3(a+b x)}{3 b}-\frac {\sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2564, 14} \[ \frac {\sin ^3(a+b x)}{3 b}-\frac {\sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \sin ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^3(a+b x)}{3 b}-\frac {\sin ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 27, normalized size = 0.87 \[ \frac {\sin ^3(a+b x) (3 \cos (2 (a+b x))+7)}{30 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 33, normalized size = 1.06 \[ -\frac {{\left (3 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 26, normalized size = 0.84 \[ -\frac {3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.29 \[ \frac {-\frac {\sin \left (b x +a \right ) \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{15}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 26, normalized size = 0.84 \[ -\frac {3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 26, normalized size = 0.84 \[ \frac {5\,{\sin \left (a+b\,x\right )}^3-3\,{\sin \left (a+b\,x\right )}^5}{15\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.02, size = 44, normalized size = 1.42 \[ \begin {cases} \frac {2 \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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