Optimal. Leaf size=11 \[ \frac{\log (a+b \tan (x))}{b} \]
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Rubi [A] time = 0.0341299, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3506, 31} \[ \frac{\log (a+b \tan (x))}{b} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{a+b \tan (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (x)\right )}{b}\\ &=\frac{\log (a+b \tan (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0652282, size = 20, normalized size = 1.82 \[ \frac{\log (a \cos (x)+b \sin (x))-\log (\cos (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953026, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \tan \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25314, size = 107, normalized size = 9.73 \begin{align*} \frac{\log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - \log \left (\cos \left (x\right )^{2}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (x \right )}}{a + b \tan{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08348, size = 16, normalized size = 1.45 \begin{align*} \frac{\log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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