Optimal. Leaf size=10 \[ \sqrt {a \sec ^2(x)} \]
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Rubi [A]
time = 0.03, 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 = {3738, 4209, 32}
\begin {gather*} \sqrt {a \sec ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3738
Rule 4209
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 \text {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 \begin {gather*} \sqrt {a \sec ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 11, normalized size = 1.10
method | result | size |
derivativedivides | \(\sqrt {a +a \left (\tan ^{2}\left (x \right )\right )}\) | \(11\) |
default | \(\sqrt {a +a \left (\tan ^{2}\left (x \right )\right )}\) | \(11\) |
risch | \(2 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\) | \(21\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.00, size = 10, normalized size = 1.00 \begin {gather*} \sqrt {a \tan \left (x\right )^{2} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 10, normalized size = 1.00 \begin {gather*} \sqrt {a \tan ^{2}{\left (x \right )} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 10, normalized size = 1.00 \begin {gather*} \sqrt {a \tan \left (x\right )^{2} + a} \end {gather*}
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 \begin {gather*} \frac {\sqrt {a}}{\sqrt {{\cos \left (x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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