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 = 0.01, antiderivative size = 36, 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, 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 8
Rule 2635
Rule 4123
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.68, size = 27, normalized size = 0.75 \[ \frac {{\left (\cos \relax (x)^{3} \sin \relax (x) + x \cos \relax (x)^{2}\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 39, normalized size = 1.08 \[ -\frac {1}{2} \, \sqrt {a} {\left (\frac {\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - x}{a} - \frac {\tan \relax (x)}{{\left (\tan \relax (x)^{2} + 1\right )} a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 22, normalized size = 0.61 \[ \frac {\cos \relax (x ) \sin \relax (x )+x}{2 \cos \relax (x )^{2} \sqrt {\frac {a}{\cos \relax (x )^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 25, normalized size = 0.69 \[ \frac {x}{2 \, \sqrt {a}} + \frac {\tan \relax (x)}{2 \, {\left (\sqrt {a} \tan \relax (x)^{2} + \sqrt {a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\frac {a}{{\cos \relax (x)}^4}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \sec ^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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