Optimal. Leaf size=46 \[ -\frac {\cos ^{12}(a+b x)}{12 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^8(a+b x)}{8 b} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2565, 266, 43} \[ -\frac {\cos ^{12}(a+b x)}{12 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^8(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2565
Rubi steps
\begin {align*} \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(a+b x)\right )}{2 b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(a+b x)\right )}{2 b}\\ &=-\frac {\cos ^8(a+b x)}{8 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^{12}(a+b x)}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 68, normalized size = 1.48 \[ -\frac {600 \cos (2 (a+b x))+75 \cos (4 (a+b x))-100 \cos (6 (a+b x))-30 \cos (8 (a+b x))+12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{122880 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 36, normalized size = 0.78 \[ -\frac {10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}}{120 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 85, normalized size = 1.85 \[ -\frac {\cos \left (12 \, b x + 12 \, a\right )}{24576 \, b} - \frac {\cos \left (10 \, b x + 10 \, a\right )}{10240 \, b} + \frac {\cos \left (8 \, b x + 8 \, a\right )}{4096 \, b} + \frac {5 \, \cos \left (6 \, b x + 6 \, a\right )}{6144 \, b} - \frac {5 \, \cos \left (4 \, b x + 4 \, a\right )}{8192 \, b} - \frac {5 \, \cos \left (2 \, b x + 2 \, a\right )}{1024 \, b} \]
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 \[ \frac {-\frac {\left (\sin ^{4}\left (b x +a \right )\right ) \left (\cos ^{8}\left (b x +a \right )\right )}{12}-\frac {\left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{8}\left (b x +a \right )\right )}{30}-\frac {\left (\cos ^{8}\left (b x +a \right )\right )}{120}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 46, normalized size = 1.00 \[ -\frac {10 \, \sin \left (b x + a\right )^{12} - 36 \, \sin \left (b x + a\right )^{10} + 45 \, \sin \left (b x + a\right )^{8} - 20 \, \sin \left (b x + a\right )^{6}}{120 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 35, normalized size = 0.76 \[ -\frac {{\cos \left (a+b\,x\right )}^8\,\left (10\,{\cos \left (a+b\,x\right )}^4-24\,{\cos \left (a+b\,x\right )}^2+15\right )}{120\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 71.14, size = 65, normalized size = 1.41 \[ \begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos ^{8}{\left (a + b x \right )}}{8 b} - \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{10}{\left (a + b x \right )}}{20 b} - \frac {\cos ^{12}{\left (a + b x \right )}}{120 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\relax (a )} \cos ^{7}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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