Optimal. Leaf size=12 \[ -\frac{1}{\sqrt{a \sec ^2(x)}} \]
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Rubi [A] time = 0.0464586, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3657, 4124, 32} \[ -\frac{1}{\sqrt{a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4124
Rule 32
Rubi steps
\begin{align*} \int \frac{\tan (x)}{\sqrt{a+a \tan ^2(x)}} \, dx &=\int \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{(a x)^{3/2}} \, dx,x,\sec ^2(x)\right )\\ &=-\frac{1}{\sqrt{a \sec ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0128336, size = 12, normalized size = 1. \[ -\frac{1}{\sqrt{a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 13, normalized size = 1.1 \begin{align*} -{\frac{1}{\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (x\right )}{\sqrt{a \tan \left (x\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37088, size = 34, normalized size = 2.83 \begin{align*} -\frac{1}{\sqrt{a \tan \left (x\right )^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.609833, size = 14, normalized size = 1.17 \begin{align*} - \frac{1}{\sqrt{a \tan ^{2}{\left (x \right )} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08739, size = 16, normalized size = 1.33 \begin{align*} -\frac{1}{\sqrt{a \tan \left (x\right )^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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