Optimal. Leaf size=13 \[ \frac{\tan (x)}{\sqrt{-\sec ^2(x)}} \]
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Rubi [A] time = 0.0198659, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 191} \[ \frac{\tan (x)}{\sqrt{-\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-\tan ^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\sec ^2(x)}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\left (-1-x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0070587, size = 13, normalized size = 1. \[ \frac{\tan (x)}{\sqrt{-\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 14, normalized size = 1.1 \begin{align*}{\tan \left ( x \right ){\frac{1}{\sqrt{-1- \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{1}{\sqrt{-\tan \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.71258, size = 42, normalized size = 3.23 \begin{align*} -\frac{1}{2} \,{\left (e^{\left (2 i \, x\right )} - 1\right )} e^{\left (-i \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \tan ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.08866, size = 16, normalized size = 1.23 \begin{align*} -\frac{i \, \tan \left (x\right )}{\sqrt{\tan \left (x\right )^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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