Optimal. Leaf size=13 \[ \frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4122, 191} \[ \frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \sec ^2(x)}} \, dx &=a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 16, normalized size = 1.23 \[ \frac {\sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) \sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 11, normalized size = 0.85 \[ \frac {\sin \relax (x)}{\sqrt {a} \mathrm {sgn}\left (\cos \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 16, normalized size = 1.23 \[ \frac {\sin \relax (x )}{\sqrt {\frac {a}{\cos \relax (x )^{2}}}\, \cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 6, normalized size = 0.46 \[ \frac {\sin \relax (x)}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 15, normalized size = 1.15 \[ \frac {\sqrt {2}\,\sin \left (2\,x\right )}{2\,\sqrt {a}\,\sqrt {2\,{\cos \relax (x)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 15, normalized size = 1.15 \[ \frac {\tan {\relax (x )}}{\sqrt {a} \sqrt {\sec ^{2}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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