Optimal. Leaf size=46 \[ -\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {\sin (a+b x) \cos (a+b x)}{8 b}+\frac {x}{8} \]
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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 = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {\sin (a+b x) \cos (a+b x)}{8 b}+\frac {x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sin ^2(a+b x) \, dx &=-\frac {\cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac {1}{4} \int \cos ^2(a+b x) \, dx\\ &=\frac {\cos (a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac {\int 1 \, dx}{8}\\ &=\frac {x}{8}+\frac {\cos (a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 0.50 \[ -\frac {\sin (4 (a+b x))-4 (a+b x)}{32 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 36, normalized size = 0.78 \[ \frac {b x - {\left (2 \, \cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 18, normalized size = 0.39 \[ \frac {1}{8} \, x - \frac {\sin \left (4 \, b x + 4 \, a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 43, normalized size = 0.93 \[ \frac {-\frac {\left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{4}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{8}+\frac {b x}{8}+\frac {a}{8}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 24, normalized size = 0.52 \[ \frac {4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 50, normalized size = 1.09 \[ \frac {x}{8}-\frac {\frac {\mathrm {tan}\left (a+b\,x\right )}{8}-\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{8}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^4+2\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.45, size = 92, normalized size = 2.00 \[ \begin {cases} \frac {x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {\sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {\sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \cos ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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