Optimal. Leaf size=13 \[ \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \]
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Rubi [A] time = 0.0286669, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
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*}
Mathematica [A] time = 0.0053793, size = 13, normalized size = 1. \[ \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 16, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }{\frac{1}{\sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.88085, size = 8, normalized size = 0.62 \begin{align*} \frac{\sin \left (x\right )}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44531, size = 46, normalized size = 3.54 \begin{align*} \frac{\sqrt{\frac{a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) \sin \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.53778, size = 15, normalized size = 1.15 \begin{align*} \frac{\tan{\left (x \right )}}{\sqrt{a} \sqrt{\sec ^{2}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22898, size = 15, normalized size = 1.15 \begin{align*} \frac{\sin \left (x\right )}{\sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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