Optimal. Leaf size=18 \[ \frac{\log (\sin (a+b x)+\cos (a+b x))}{b} \]
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Rubi [A] time = 0.0281176, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {3133} \[ \frac{\log (\sin (a+b x)+\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3133
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)-\sin (a+b x)}{\cos (a+b x)+\sin (a+b x)} \, dx &=\frac{\log (\cos (a+b x)+\sin (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.043197, size = 18, normalized size = 1. \[ \frac{\log (\sin (a+b x)+\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936929, size = 24, normalized size = 1.33 \begin{align*} \frac{\log \left (\cos \left (b x + a\right ) + \sin \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05172, size = 59, normalized size = 3.28 \begin{align*} \frac{\log \left (2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.484454, size = 31, normalized size = 1.72 \begin{align*} \begin{cases} \frac{\log{\left (\sin{\left (a + b x \right )} + \cos{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x \left (- \sin{\left (a \right )} + \cos{\left (a \right )}\right )}{\sin{\left (a \right )} + \cos{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13265, size = 39, normalized size = 2.17 \begin{align*} -\frac{\log \left (\tan \left (b x + a\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (b x + a\right ) + 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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