Optimal. Leaf size=11 \[ \frac {\log (a+b \tan (x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 31
Rule 3506
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 1.82, size = 40, normalized size = 3.64 \[ \frac {\log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - \log \left (\cos \relax (x)^{2}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 12, normalized size = 1.09 \[ \frac {\log \left ({\left | b \tan \relax (x) + a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \tan \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \tan \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 11, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\mathrm {tan}\relax (x)\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\relax (x )}}{a + b \tan {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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