Optimal. Leaf size=13 \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (5-x^2\right )\right ) \]
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Rubi [A] time = 0.0117344, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4204, 3770} \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (5-x^2\right )\right ) \]
Antiderivative was successfully verified.
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Rule 4204
Rule 3770
Rubi steps
\begin{align*} \int x \sec \left (5-x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sec (5-x) \, dx,x,x^2\right )\\ &=-\frac{1}{2} \tanh ^{-1}\left (\sin \left (5-x^2\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0175405, size = 13, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (5-x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( \sec \left ({x}^{2}-5 \right ) +\tan \left ({x}^{2}-5 \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962089, size = 22, normalized size = 1.69 \begin{align*} \frac{1}{2} \, \log \left (\sec \left (x^{2} - 5\right ) + \tan \left (x^{2} - 5\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.32914, size = 76, normalized size = 5.85 \begin{align*} \frac{1}{4} \, \log \left (\sin \left (x^{2} - 5\right ) + 1\right ) - \frac{1}{4} \, \log \left (-\sin \left (x^{2} - 5\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.97906, size = 15, normalized size = 1.15 \begin{align*} \frac{\log{\left (\tan{\left (x^{2} - 5 \right )} + \sec{\left (x^{2} - 5 \right )} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10437, size = 55, normalized size = 4.23 \begin{align*} \frac{1}{8} \, \log \left ({\left | \frac{1}{\sin \left (x^{2} - 5\right )} + \sin \left (x^{2} - 5\right ) + 2 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | \frac{1}{\sin \left (x^{2} - 5\right )} + \sin \left (x^{2} - 5\right ) - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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