Optimal. Leaf size=11 \[ \frac {\log (a+b \sec (x))}{b} \]
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Rubi [A] time = 0.05, 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 29
Rule 31
Rule 36
Rule 4339
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.67, size = 19, normalized size = 1.73 \[ \frac {\log \left (a \cos \relax (x) + b\right ) - \log \left (-\cos \relax (x)\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 22, normalized size = 2.00 \[ \frac {\log \left ({\left | a \cos \relax (x) + b \right |}\right )}{b} - \frac {\log \left ({\left | \cos \relax (x) \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \sec \relax (x )\right )}{b} \]
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 \[ \frac {\log \left (b \sec \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.25, size = 48, normalized size = 4.36 \[ \frac {\mathrm {atan}\left (\frac {b\,{\sin \left (\frac {x}{2}\right )}^2}{a\,{\cos \left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}+b\,{\cos \left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}-a\,{\sin \left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 14, normalized size = 1.27 \[ \begin {cases} \frac {\log {\left (\frac {a}{b} + \sec {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\sec {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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