Optimal. Leaf size=11 \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {203} \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 203
Rubi steps
\begin {align*} \int \frac {1}{b^2 \cos ^2(x)+\sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 11, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\tan (x)}{b}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.04, size = 31, normalized size = 2.82 \[ -\frac {\arctan \left (\frac {{\left (b^{2} + 1\right )} \cos \relax (x)^{2} - 1}{2 \, b \cos \relax (x) \sin \relax (x)}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 22, normalized size = 2.00 \[ \frac {\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \relax (x)}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 12, normalized size = 1.09 \[ \frac {\arctan \left (\frac {\tan \relax (x )}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 11, normalized size = 1.00 \[ \frac {\arctan \left (\frac {\tan \relax (x)}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.83, size = 11, normalized size = 1.00 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\relax (x)}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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