Optimal. Leaf size=46 \[ -\frac {\cos ^9(a+b x)}{9 b}+\frac {2 \cos ^7(a+b x)}{7 b}-\frac {\cos ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 46, 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, 270} \[ -\frac {\cos ^9(a+b x)}{9 b}+\frac {2 \cos ^7(a+b x)}{7 b}-\frac {\cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rubi steps
\begin {align*} \int \cos ^4(a+b x) \sin ^5(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^5(a+b x)}{5 b}+\frac {2 \cos ^7(a+b x)}{7 b}-\frac {\cos ^9(a+b x)}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 37, normalized size = 0.80 \[ \frac {\cos ^5(a+b x) (220 \cos (2 (a+b x))-35 \cos (4 (a+b x))-249)}{2520 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 36, normalized size = 0.78 \[ -\frac {35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 68, normalized size = 1.48 \[ -\frac {\cos \left (9 \, b x + 9 \, a\right )}{2304 \, b} + \frac {\cos \left (7 \, b x + 7 \, a\right )}{1792 \, b} + \frac {\cos \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac {\cos \left (3 \, b x + 3 \, a\right )}{192 \, b} - \frac {3 \, \cos \left (b x + a\right )}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 1.13 \[ \frac {-\frac {\left (\cos ^{5}\left (b x +a \right )\right ) \left (\sin ^{4}\left (b x +a \right )\right )}{9}-\frac {4 \left (\cos ^{5}\left (b x +a \right )\right ) \left (\sin ^{2}\left (b x +a \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (b x +a \right )\right )}{315}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 36, normalized size = 0.78 \[ -\frac {35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 36, normalized size = 0.78 \[ -\frac {35\,{\cos \left (a+b\,x\right )}^9-90\,{\cos \left (a+b\,x\right )}^7+63\,{\cos \left (a+b\,x\right )}^5}{315\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.09, size = 68, normalized size = 1.48 \[ \begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{35 b} - \frac {8 \cos ^{9}{\left (a + b x \right )}}{315 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\relax (a )} \cos ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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