Optimal. Leaf size=11 \[ \frac{\log (a+b \sec (x))}{b} \]
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Rubi [A] time = 0.0456383, antiderivative size = 20, normalized size of antiderivative = 1.82, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {4339, 36, 29, 31} \[ \frac{\log (a \cos (x)+b)}{b}-\frac{\log (\cos (x))}{b} \]
Antiderivative was successfully verified.
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Rule 4339
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec (x) \tan (x)}{a+b \sec (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x (b+a x)} \, dx,x,\cos (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\cos (x)\right )}{b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{b+a x} \, dx,x,\cos (x)\right )}{b}\\ &=-\frac{\log (\cos (x))}{b}+\frac{\log (b+a \cos (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0176336, size = 20, normalized size = 1.82 \[ \frac{\log (a \cos (x)+b)}{b}-\frac{\log (\cos (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\sec \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960017, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \sec \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48189, size = 51, normalized size = 4.64 \begin{align*} \frac{\log \left (a \cos \left (x\right ) + b\right ) - \log \left (-\cos \left (x\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.572914, size = 14, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\log{\left (\frac{a}{b} + \sec{\left (x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{\sec{\left (x \right )}}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10937, size = 30, normalized size = 2.73 \begin{align*} \frac{\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{b} - \frac{\log \left ({\left | \cos \left (x\right ) \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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