Optimal. Leaf size=88 \[ -\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac {\sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac {5 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac {5 \sin (a+b x) \cos (a+b x)}{128 b}+\frac {5 x}{128} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac {\sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac {5 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac {5 \sin (a+b x) \cos (a+b x)}{128 b}+\frac {5 x}{128} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx &=-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {1}{8} \int \cos ^6(a+b x) \, dx\\ &=\frac {\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {5}{48} \int \cos ^4(a+b x) \, dx\\ &=\frac {5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {5}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac {5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {5 \int 1 \, dx}{128}\\ &=\frac {5 x}{128}+\frac {5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 52, normalized size = 0.59 \[ \frac {48 \sin (2 (a+b x))-24 \sin (4 (a+b x))-16 \sin (6 (a+b x))-3 \sin (8 (a+b x))+120 b x}{3072 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 57, normalized size = 0.65 \[ \frac {15 \, b x - {\left (48 \, \cos \left (b x + a\right )^{7} - 8 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} - 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 60, normalized size = 0.68 \[ \frac {5}{128} \, x - \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 64, normalized size = 0.73 \[ \frac {-\frac {\sin \left (b x +a \right ) \left (\cos ^{7}\left (b x +a \right )\right )}{8}+\frac {\left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{48}+\frac {5 b x}{128}+\frac {5 a}{128}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 48, normalized size = 0.55 \[ \frac {64 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 120 \, b x + 120 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 24 \, \sin \left (4 \, b x + 4 \, a\right )}{3072 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 89, normalized size = 1.01 \[ \frac {5\,x}{128}+\frac {\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {55\,{\mathrm {tan}\left (a+b\,x\right )}^5}{384}+\frac {73\,{\mathrm {tan}\left (a+b\,x\right )}^3}{384}-\frac {5\,\mathrm {tan}\left (a+b\,x\right )}{128}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+4\,{\mathrm {tan}\left (a+b\,x\right )}^6+6\,{\mathrm {tan}\left (a+b\,x\right )}^4+4\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.10, size = 189, normalized size = 2.15 \[ \begin {cases} \frac {5 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {5 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {15 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {5 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {5 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac {5 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} + \frac {55 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac {73 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac {5 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \cos ^{6}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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