Optimal. Leaf size=8 \[ \frac {\sec \left (x^2\right )}{2} \]
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Rubi [A] time = 0.06, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6715, 2606, 8} \[ \frac {\sec \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 6715
Rubi steps
\begin {align*} \int x \sec \left (x^2\right ) \tan \left (x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \sec (x) \tan (x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int 1 \, dx,x,\sec \left (x^2\right )\right )\\ &=\frac {\sec \left (x^2\right )}{2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 8, normalized size = 1.00 \[ \frac {\sec \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 8, normalized size = 1.00 \[ \frac {1}{2 \, \cos \left (x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 8, normalized size = 1.00 \[ \frac {1}{2 \, \cos \left (x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 7, normalized size = 0.88 \[ \frac {\sec \left (x^{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 56, normalized size = 7.00 \[ \frac {\cos \left (2 \, x^{2}\right ) \cos \left (x^{2}\right ) + \sin \left (2 \, x^{2}\right ) \sin \left (x^{2}\right ) + \cos \left (x^{2}\right )}{\cos \left (2 \, x^{2}\right )^{2} + \sin \left (2 \, x^{2}\right )^{2} + 2 \, \cos \left (2 \, x^{2}\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 8, normalized size = 1.00 \[ \frac {1}{2\,\cos \left (x^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 5, normalized size = 0.62 \[ \frac {\sec {\left (x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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