Optimal. Leaf size=16 \[ \frac{\sin (a+b x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.0541188, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {4380, 2635, 8} \[ \frac{\sin (a+b x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4380
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^2(a+b x)-\sin ^2(a+b x)}{\cos ^2(a+b x)+\sin ^2(a+b x)} \, dx &=\int \left (\cos ^2(a+b x)-\sin ^2(a+b x)\right ) \, dx\\ &=\int \cos ^2(a+b x) \, dx-\int \sin ^2(a+b x) \, dx\\ &=\frac{\cos (a+b x) \sin (a+b x)}{b}\\ \end{align*}
Mathematica [B] time = 0.0118323, size = 33, normalized size = 2.06 \[ \frac{\sin (2 a) \cos (2 b x)}{2 b}+\frac{\cos (2 a) \sin (2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 17, normalized size = 1.1 \begin{align*}{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951656, size = 30, normalized size = 1.88 \begin{align*} \frac{\tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99053, size = 39, normalized size = 2.44 \begin{align*} \frac{\cos \left (b x + a\right ) \sin \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.283215, size = 32, normalized size = 2. \begin{align*} \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b \sin ^{2}{\left (a + b x \right )} + b \cos ^{2}{\left (a + b x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08693, size = 19, normalized size = 1.19 \begin{align*} \frac{\sin \left (2 \, b x + 2 \, a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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