Optimal. Leaf size=3 \[ \sinh ^{-1}(\tan (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 3, 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 = {3738, 4207,
221} \begin {gather*} \sinh ^{-1}(\tan (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 3738
Rule 4207
Rubi steps
\begin {align*} \int \sqrt {1+\tan ^2(x)} \, dx &=\int \sqrt {\sec ^2(x)} \, dx\\ &=\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tan (x)\right )\\ &=\sinh ^{-1}(\tan (x))\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(3)=6\).
time = 0.01, size = 44, normalized size = 14.67 \begin {gather*} \cos (x) \left (-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right ) \sqrt {\sec ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 4, normalized size = 1.33
method | result | size |
derivativedivides | \(\arcsinh \left (\tan \left (x \right )\right )\) | \(4\) |
default | \(\arcsinh \left (\tan \left (x \right )\right )\) | \(4\) |
risch | \(2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )-2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 3, normalized size = 1.00 \begin {gather*} \operatorname {arsinh}\left (\tan \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (3) = 6\).
time = 1.88, size = 60, normalized size = 20.00 \begin {gather*} \frac {1}{2} \, \log \left (\frac {\tan \left (x\right )^{2} + \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {\tan \left (x\right )^{2} - \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {\tan ^{2}{\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs.
\(2 (3) = 6\).
time = 0.46, size = 16, normalized size = 5.33 \begin {gather*} -\log \left (\sqrt {\tan \left (x\right )^{2} + 1} - \tan \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 3, normalized size = 1.00 \begin {gather*} \mathrm {asinh}\left (\mathrm {tan}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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