Optimal. Leaf size=8 \[ 2 \sinh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 8, 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 = {1978, 56, 221}
\begin {gather*} 2 \sinh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 221
Rule 1978
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {x}{1+x}}}{x} \, dx &=\int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=2 \sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.75 \begin {gather*} 2 \tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs.
\(2(6)=12\).
time = 0.06, size = 32, normalized size = 4.00
method | result | size |
default | \(\frac {\sqrt {\frac {x}{1+x}}\, \left (1+x \right ) \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {x \left (1+x \right )}}\) | \(32\) |
trager | \(-\ln \left (2 \sqrt {\frac {x}{1+x}}\, x +2 \sqrt {\frac {x}{1+x}}-2 x -1\right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (6) = 12\).
time = 0.28, size = 27, normalized size = 3.38 \begin {gather*} \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x}{x + 1}} - 1\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. 27 vs.
\(2 (6) = 12\).
time = 0.35, size = 27, normalized size = 3.38 \begin {gather*} \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x}{x + 1}} - 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 \frac {\sqrt {\frac {x}{x + 1}}}{x}\, 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. 22 vs.
\(2 (6) = 12\).
time = 2.34, size = 22, normalized size = 2.75 \begin {gather*} -\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \mathrm {sgn}\left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 12, normalized size = 1.50 \begin {gather*} 2\,\mathrm {atanh}\left (\sqrt {\frac {x}{x+1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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