Optimal. Leaf size=11 \[ \frac {\log (a+b \sin (x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 31
Rule 2668
Rule 3159
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*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \sin \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 12, normalized size = 1.09 \[ \frac {\log \left ({\left | b \sin \relax (x) + a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \sin \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 50, normalized size = 4.55 \[ \frac {\log \left (a + \frac {2 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{b} - \frac {\log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 55, normalized size = 5.00 \[ \frac {2\,\mathrm {atanh}\left (\frac {b\,\left (2\,a^3\,\sin \relax (x)+\frac {5\,a^2\,b}{2}-b^3-\frac {a^2\,b\,\cos \left (2\,x\right )}{2}\right )}{{\left (2\,a^2+\sin \relax (x)\,a\,b-b^2\right )}^2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 32, normalized size = 2.91 \[ \begin {cases} \frac {\log {\left (\frac {a \sec {\relax (x )}}{b} + \tan {\relax (x )} \right )}}{b} - \frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {\tan {\relax (x )}}{a \sec {\relax (x )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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