Optimal. Leaf size=16 \[ -\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right ) \]
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Rubi [A] time = 0.0173641, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3657, 4122, 217, 203} \[ -\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \sqrt{-1-\tan ^2(x)} \, dx &=\int \sqrt{-\sec ^2(x)} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\tan (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right )\\ &=-\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right )\\ \end{align*}
Mathematica [B] time = 0.0069509, size = 46, normalized size = 2.88 \[ \cos (x) \sqrt{-\sec ^2(x)} \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 17, normalized size = 1.1 \begin{align*} -\arctan \left ({\tan \left ( x \right ){\frac{1}{\sqrt{-1- \left ( \tan \left ( x \right ) \right ) ^{2}}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.84511, size = 23, normalized size = 1.44 \begin{align*} \arctan \left (\cos \left (x\right ), \sin \left (x\right ) + 1\right ) + \arctan \left (\cos \left (x\right ), -\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.87246, size = 55, normalized size = 3.44 \begin{align*} i \, \log \left (e^{\left (i \, x\right )} + i\right ) - i \, \log \left (e^{\left (i \, x\right )} - i\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 \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 = 1.09306, size = 9, normalized size = 0.56 \begin{align*} i \, \arcsin \left (i \, \tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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