Optimal. Leaf size=31 \[ \frac {\cos ^5(a+b x)}{5 b}-\frac {\cos ^3(a+b x)}{3 b} \]
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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 = {2565, 14} \[ \frac {\cos ^5(a+b x)}{5 b}-\frac {\cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2565
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sin ^3(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^3(a+b x)}{3 b}+\frac {\cos ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 0.87 \[ \frac {\cos ^3(a+b x) (3 \cos (2 (a+b x))-7)}{30 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 26, normalized size = 0.84 \[ \frac {3 \, \cos \left (b x + a\right )^{5} - 5 \, \cos \left (b x + a\right )^{3}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 27, normalized size = 0.87 \[ \frac {\cos \left (b x + a\right )^{5}}{5 \, b} - \frac {\cos \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 34, normalized size = 1.10 \[ \frac {-\frac {\left (\cos ^{3}\left (b x +a \right )\right ) \left (\sin ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (b x +a \right )\right )}{15}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 26, normalized size = 0.84 \[ \frac {3 \, \cos \left (b x + a\right )^{5} - 5 \, \cos \left (b x + a\right )^{3}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 26, normalized size = 0.84 \[ -\frac {5\,{\cos \left (a+b\,x\right )}^3-3\,{\cos \left (a+b\,x\right )}^5}{15\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.46, size = 46, normalized size = 1.48 \[ \begin {cases} - \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 \cos ^{5}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\relax (a )} \cos ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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