Optimal. Leaf size=15 \[ \sin (x) \cos (x) \sqrt{a \sec ^4(x)} \]
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Rubi [A] time = 0.0158969, antiderivative size = 15, 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, 3767, 8} \[ \sin (x) \cos (x) \sqrt{a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sqrt{a \sec ^4(x)} \, dx &=\left (\cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \sec ^2(x) \, dx\\ &=-\left (\left (\cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (x))\right )\\ &=\cos (x) \sqrt{a \sec ^4(x)} \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0054477, size = 15, normalized size = 1. \[ \sin (x) \cos (x) \sqrt{a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 14, normalized size = 0.9 \begin{align*} \cos \left ( x \right ) \sin \left ( x \right ) \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 = 1.73949, size = 8, normalized size = 0.53 \begin{align*} \sqrt{a} \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36385, size = 43, normalized size = 2.87 \begin{align*} \sqrt{\frac{a}{\cos \left (x\right )^{4}}} \cos \left (x\right ) \sin \left (x\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 \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 = 1.2576, size = 8, normalized size = 0.53 \begin{align*} \sqrt{a} \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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