Optimal. Leaf size=15 \[ \sin (x) \cos (x) \sqrt {a \sec ^4(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4123, 3767, 8} \[ \sin (x) \cos (x) \sqrt {a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 4123
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*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \sin (x) \cos (x) \sqrt {a \sec ^4(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 13, normalized size = 0.87 \[ \sqrt {\frac {a}{\cos \relax (x)^{4}}} \cos \relax (x) \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 6, normalized size = 0.40 \[ \sqrt {a} \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 14, normalized size = 0.93 \[ \cos \relax (x ) \sin \relax (x ) \sqrt {\frac {a}{\cos \relax (x )^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 6, normalized size = 0.40 \[ \sqrt {a} \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 6, normalized size = 0.40 \[ \sqrt {a}\,\mathrm {tan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sec ^{4}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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