Optimal. Leaf size=36 \[ \frac{1}{2} \tan (x) \sqrt{a \cos ^4(x)}+\frac{1}{2} x \sec ^2(x) \sqrt{a \cos ^4(x)} \]
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Rubi [A] time = 0.0153305, 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 = {3207, 2635, 8} \[ \frac{1}{2} \tan (x) \sqrt{a \cos ^4(x)}+\frac{1}{2} x \sec ^2(x) \sqrt{a \cos ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sqrt{a \cos ^4(x)} \, dx &=\left (\sqrt{a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^2(x) \, dx\\ &=\frac{1}{2} \sqrt{a \cos ^4(x)} \tan (x)+\frac{1}{2} \left (\sqrt{a \cos ^4(x)} \sec ^2(x)\right ) \int 1 \, dx\\ &=\frac{1}{2} x \sqrt{a \cos ^4(x)} \sec ^2(x)+\frac{1}{2} \sqrt{a \cos ^4(x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0137499, size = 25, normalized size = 0.69 \[ \frac{1}{2} \sec ^2(x) \sqrt{a \cos ^4(x)} (x+\sin (x) \cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.192, 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}}\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 = 2.4277, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{2} \, \sqrt{a} x + \frac{\sqrt{a} \tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04887, size = 69, normalized size = 1.92 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4}}{\left (\cos \left (x\right ) \sin \left (x\right ) + x\right )}}{2 \, \cos \left (x\right )^{2}} \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 [A] time = 1.55724, size = 18, normalized size = 0.5 \begin{align*} \frac{1}{4} \, \sqrt{a}{\left (2 \, x + \sin \left (2 \, x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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