Optimal. Leaf size=36 \[ \frac{x \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}+\frac{\tan (x)}{2 \sqrt{a \sec ^4(x)}} \]
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Rubi [A] time = 0.0146596, antiderivative size = 36, 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 = {4123, 2635, 8} \[ \frac{x \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}+\frac{\tan (x)}{2 \sqrt{a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sec ^4(x)}} \, dx &=\frac{\sec ^2(x) \int \cos ^2(x) \, dx}{\sqrt{a \sec ^4(x)}}\\ &=\frac{\tan (x)}{2 \sqrt{a \sec ^4(x)}}+\frac{\sec ^2(x) \int 1 \, dx}{2 \sqrt{a \sec ^4(x)}}\\ &=\frac{x \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}+\frac{\tan (x)}{2 \sqrt{a \sec ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0234295, size = 23, normalized size = 0.64 \[ \frac{\tan (x)+x \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 22, normalized size = 0.6 \begin{align*}{\frac{\cos \left ( x \right ) \sin \left ( x \right ) +x}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{4}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66388, size = 34, normalized size = 0.94 \begin{align*} \frac{x}{2 \, \sqrt{a}} + \frac{\tan \left (x\right )}{2 \,{\left (\sqrt{a} \tan \left (x\right )^{2} + \sqrt{a}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4198, size = 74, normalized size = 2.06 \begin{align*} \frac{{\left (\cos \left (x\right )^{3} \sin \left (x\right ) + x \cos \left (x\right )^{2}\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4}}}}{2 \, a} \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 \frac{1}{\sqrt{a \sec ^{4}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31144, size = 53, normalized size = 1.47 \begin{align*} -\frac{1}{2} \, \sqrt{a}{\left (\frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor - x}{a} - \frac{\tan \left (x\right )}{{\left (\tan \left (x\right )^{2} + 1\right )} a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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