Optimal. Leaf size=46 \[ -\frac {\cos ^{11}(a+b x)}{11 b}+\frac {2 \cos ^9(a+b x)}{9 b}-\frac {\cos ^7(a+b x)}{7 b} \]
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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} \[ -\frac {\cos ^{11}(a+b x)}{11 b}+\frac {2 \cos ^9(a+b x)}{9 b}-\frac {\cos ^7(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rubi steps
\begin {align*} \int \cos ^6(a+b x) \sin ^5(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^7(a+b x)}{7 b}+\frac {2 \cos ^9(a+b x)}{9 b}-\frac {\cos ^{11}(a+b x)}{11 b}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 37, normalized size = 0.80 \[ \frac {\cos ^7(a+b x) (364 \cos (2 (a+b x))-63 \cos (4 (a+b x))-365)}{5544 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 36, normalized size = 0.78 \[ -\frac {63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}}{693 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 82, normalized size = 1.78 \[ -\frac {\cos \left (11 \, b x + 11 \, a\right )}{11264 \, b} - \frac {\cos \left (9 \, b x + 9 \, a\right )}{9216 \, b} + \frac {5 \, \cos \left (7 \, b x + 7 \, a\right )}{7168 \, b} + \frac {\cos \left (5 \, b x + 5 \, a\right )}{1024 \, b} - \frac {5 \, \cos \left (3 \, b x + 3 \, a\right )}{1536 \, b} - \frac {5 \, \cos \left (b x + a\right )}{512 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 52, normalized size = 1.13 \[ \frac {-\frac {\left (\cos ^{7}\left (b x +a \right )\right ) \left (\sin ^{4}\left (b x +a \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (b x +a \right )\right ) \left (\sin ^{2}\left (b x +a \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (b x +a \right )\right )}{693}}{b} \]
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 \[ -\frac {63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}}{693 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 36, normalized size = 0.78 \[ -\frac {63\,{\cos \left (a+b\,x\right )}^{11}-154\,{\cos \left (a+b\,x\right )}^9+99\,{\cos \left (a+b\,x\right )}^7}{693\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 47.18, size = 68, normalized size = 1.48 \[ \begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{7 b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \cos ^{9}{\left (a + b x \right )}}{63 b} - \frac {8 \cos ^{11}{\left (a + b x \right )}}{693 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\relax (a )} \cos ^{6}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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