Optimal. Leaf size=11 \[ \frac{\tan (x)}{\sqrt{\sec ^2(x)}} \]
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Rubi [A] time = 0.0073548, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4122, 191} \[ \frac{\tan (x)}{\sqrt{\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
Rubi steps
\begin{align*} \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 = 0.0058971, size = 11, normalized size = 1. \[ \frac{\tan (x)}{\sqrt{\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 14, normalized size = 1.3 \begin{align*}{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }{\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{-2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18009, size = 15, normalized size = 1.36 \begin{align*} \frac{\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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Fricas [A] time = 1.2399, size = 12, normalized size = 1.09 \begin{align*} -\sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.482606, size = 10, normalized size = 0.91 \begin{align*} \frac{\tan{\left (x \right )}}{\sqrt{\sec ^{2}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27162, size = 8, normalized size = 0.73 \begin{align*} \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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