Optimal. Leaf size=13 \[ \frac {\tan (x)}{\sqrt {-\sec ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3657, 4122, 191} \[ \frac {\tan (x)}{\sqrt {-\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3657
Rule 4122
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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\tan (x)}{\sqrt {-\sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.39, size = 12, normalized size = 0.92 \[ -\frac {1}{2} \, {\left (e^{\left (2 i \, x\right )} - 1\right )} e^{\left (-i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.35, size = 12, normalized size = 0.92 \[ -\frac {i \, \tan \relax (x)}{\sqrt {\tan \relax (x)^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 14, normalized size = 1.08 \[ \frac {\tan \relax (x )}{\sqrt {-1-\left (\tan ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\tan \relax (x)^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.98, size = 13, normalized size = 1.00 \[ -\frac {\sqrt {2}\,\sin \left (2\,x\right )\,1{}\mathrm {i}}{2\,\sqrt {2\,{\cos \relax (x)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \tan ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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