Optimal. Leaf size=13 \[ \sqrt {\cos ^2(x)+1} \sec (x) \]
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Rubi [A] time = 0.08, antiderivative size = 13, 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 = {264} \[ \sqrt {\cos ^2(x)+1} \sec (x) \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {\sec (x) \tan (x)}{\sqrt {1+\cos ^2(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^2}} \, dx,x,\cos (x)\right )\\ &=\sqrt {1+\cos ^2(x)} \sec (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 13, normalized size = 1.00 \[ \sqrt {\cos ^2(x)+1} \sec (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 16, normalized size = 1.23 \[ \frac {\sqrt {\cos \relax (x)^{2} + 1} + \cos \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 21, normalized size = 1.62 \[ -\frac {2}{{\left (\sqrt {\cos \relax (x)^{2} + 1} - \cos \relax (x)\right )}^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 25, normalized size = 1.92 \[ \frac {1+\sec ^{2}\relax (x )}{\sqrt {\frac {1+\sec ^{2}\relax (x )}{\sec \relax (x )^{2}}}\, \sec \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 13, normalized size = 1.00 \[ \frac {\sqrt {\cos \relax (x)^{2} + 1}}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 13, normalized size = 1.00 \[ \frac {\sqrt {{\cos \relax (x)}^2+1}}{\cos \relax (x)} \]
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 \[ \int \frac {\tan {\relax (x )} \sec {\relax (x )}}{\sqrt {\cos ^{2}{\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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