Optimal. Leaf size=13 \[ -\frac {1}{2} \tanh ^{-1}\left (\sin \left (5-x^2\right )\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 3770
Rule 4204
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*}
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Mathematica [A] time = 0.02, size = 13, normalized size = 1.00 \[ -\frac {1}{2} \tanh ^{-1}\left (\sin \left (5-x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 25, normalized size = 1.92 \[ \frac {1}{4} \, \log \left (\sin \left (x^{2} - 5\right ) + 1\right ) - \frac {1}{4} \, \log \left (-\sin \left (x^{2} - 5\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 41, normalized size = 3.15 \[ \frac {1}{8} \, \log \left ({\left | \frac {1}{\sin \left (x^{2} - 5\right )} + \sin \left (x^{2} - 5\right ) + 2 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | \frac {1}{\sin \left (x^{2} - 5\right )} + \sin \left (x^{2} - 5\right ) - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 1.31 \[ \frac {\ln \left (\sec \left (x^{2}-5\right )+\tan \left (x^{2}-5\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 16, normalized size = 1.23 \[ \frac {1}{2} \, \log \left (\sec \left (x^{2} - 5\right ) + \tan \left (x^{2} - 5\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 15, normalized size = 1.15 \[ -\mathrm {atan}\left ({\mathrm {e}}^{-5{}\mathrm {i}}\,{\mathrm {e}}^{x^2\,1{}\mathrm {i}}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 15, normalized size = 1.15 \[ \frac {\log {\left (\tan {\left (x^{2} - 5 \right )} + \sec {\left (x^{2} - 5 \right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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