Optimal. Leaf size=11 \[ \frac{\log (a+b \sin (x))}{b} \]
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Rubi [A] time = 0.0342235, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3159, 2668, 31} \[ \frac{\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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Rule 3159
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a \sec (x)+b \tan (x)} \, dx &=\int \frac{\cos (x)}{a+b \sin (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sin (x)\right )}{b}\\ &=\frac{\log (a+b \sin (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0064317, size = 11, normalized size = 1. \[ \frac{\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\sin \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53328, size = 68, normalized size = 6.18 \begin{align*} \frac{\log \left (a + \frac{2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{b} - \frac{\log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28446, size = 28, normalized size = 2.55 \begin{align*} \frac{\log \left (b \sin \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.59975, size = 32, normalized size = 2.91 \begin{align*} \begin{cases} \frac{\log{\left (\frac{a \sec{\left (x \right )}}{b} + \tan{\left (x \right )} \right )}}{b} - \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2 b} & \text{for}\: b \neq 0 \\\frac{\tan{\left (x \right )}}{a \sec{\left (x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13482, size = 16, normalized size = 1.45 \begin{align*} \frac{\log \left ({\left | b \sin \left (x\right ) + a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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