Optimal. Leaf size=8 \[ \frac{\tan \left (x^2\right )}{2} \]
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Rubi [A] time = 0.0130906, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4204, 3767, 8} \[ \frac{\tan \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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Rule 4204
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x \sec ^2\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sec ^2(x) \, dx,x,x^2\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int 1 \, dx,x,-\tan \left (x^2\right )\right )\right )\\ &=\frac{\tan \left (x^2\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0163709, size = 8, normalized size = 1. \[ \frac{\tan \left (x^2\right )}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 7, normalized size = 0.9 \begin{align*}{\frac{\tan \left ({x}^{2} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.964929, size = 47, normalized size = 5.88 \begin{align*} \frac{\sin \left (2 \, 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97484, size = 31, normalized size = 3.88 \begin{align*} \frac{\sin \left (x^{2}\right )}{2 \, \cos \left (x^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sec ^{2}{\left (x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07706, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \tan \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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