Optimal. Leaf size=10 \[ \sqrt {a \sec ^2(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3657, 4124, 32} \[ \sqrt {a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3657
Rule 4124
Rubi steps
\begin {align*} \int \tan (x) \sqrt {a+a \tan ^2(x)} \, dx &=\int \sqrt {a \sec ^2(x)} \tan (x) \, dx\\ &=\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {a x}} \, dx,x,\sec ^2(x)\right )\\ &=\sqrt {a \sec ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \sqrt {a \sec ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 10, normalized size = 1.00 \[ \sqrt {a \tan \relax (x)^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 10, normalized size = 1.00 \[ \sqrt {a \tan \relax (x)^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 11, normalized size = 1.10 \[ \sqrt {a +a \left (\tan ^{2}\relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \tan \relax (x)^{2} + a} \tan \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.63, size = 10, normalized size = 1.00 \[ \frac {\sqrt {a}}{\sqrt {{\cos \relax (x)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 10, normalized size = 1.00 \[ \sqrt {a \tan ^{2}{\relax (x )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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