Optimal. Leaf size=15 \[ \frac{\sin (x) \cos (x)}{\sqrt{a \cos ^4(x)}} \]
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Rubi [A] time = 0.0140777, 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 = {3207, 3767, 8} \[ \frac{\sin (x) \cos (x)}{\sqrt{a \cos ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cos ^4(x)}} \, dx &=\frac{\cos ^2(x) \int \sec ^2(x) \, dx}{\sqrt{a \cos ^4(x)}}\\ &=-\frac{\cos ^2(x) \operatorname{Subst}(\int 1 \, dx,x,-\tan (x))}{\sqrt{a \cos ^4(x)}}\\ &=\frac{\cos (x) \sin (x)}{\sqrt{a \cos ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0055221, size = 15, normalized size = 1. \[ \frac{\sin (x) \cos (x)}{\sqrt{a \cos ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 14, normalized size = 0.9 \begin{align*}{\cos \left ( x \right ) \sin \left ( x \right ){\frac{1}{\sqrt{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 = 2.05925, size = 8, normalized size = 0.53 \begin{align*} \frac{\tan \left (x\right )}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02808, size = 51, normalized size = 3.4 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4}} \sin \left (x\right )}{a \cos \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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