Optimal. Leaf size=11 \[ \frac{\tan (x)}{\sqrt{\sec ^2(x)}} \]
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Rubi [A] time = 0.0128562, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 = 0.0067172, size = 11, normalized size = 1. \[ \frac{\tan (x)}{\sqrt{\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 12, 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 [A] time = 1.10782, 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 = 1.86436, size = 36, normalized size = 3.27 \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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Sympy [A] time = 0.422575, size = 12, normalized size = 1.09 \begin{align*} \frac{\tan{\left (x \right )}}{\sqrt{\tan ^{2}{\left (x \right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09879, 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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