Optimal. Leaf size=10 \[ \sqrt{a \sec ^2(x)} \]
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Rubi [A] time = 0.0431939, antiderivative size = 10, 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} \[ \sqrt{a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4124
Rule 32
Rubi steps
\begin{align*} \int \tan (x) \sqrt{a+a \tan ^2(x)} \, dx &=\int \sqrt{a \sec ^2(x)} \tan (x) \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a x}} \, dx,x,\sec ^2(x)\right )\\ &=\sqrt{a \sec ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0088662, size = 10, normalized size = 1. \[ \sqrt{a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 11, normalized size = 1.1 \begin{align*} \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 \sqrt{a \tan \left (x\right )^{2} + a} \tan \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31103, size = 30, normalized size = 3. \begin{align*} \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 = 0.671388, size = 10, normalized size = 1. \begin{align*} \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 = 1.1183, size = 14, normalized size = 1.4 \begin{align*} \sqrt{a \tan \left (x\right )^{2} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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