Optimal. Leaf size=61 \[ \frac {\left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}} \sinh ^{-1}(x)}{1+x} \]
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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6855, 984, 655,
221} \begin {gather*} \frac {\sqrt {\frac {2 x}{x^2+1}+1} \left (x^2+1\right )}{x+1}+\frac {\sqrt {\frac {2 x}{x^2+1}+1} \sqrt {x^2+1} \sinh ^{-1}(x)}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 655
Rule 984
Rule 6855
Rubi steps
\begin {align*} \int \sqrt {1+\frac {2 x}{1+x^2}} \, dx &=\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {\sqrt {1+2 x+x^2}}{\sqrt {1+x^2}} \, dx}{\sqrt {1+2 x+x^2}}\\ &=\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {2+2 x}{\sqrt {1+x^2}} \, dx}{2+2 x}\\ &=\frac {\left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {\left (2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2+2 x}\\ &=\frac {\left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}} \sinh ^{-1}(x)}{1+x}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 50, normalized size = 0.82 \begin {gather*} \frac {\sqrt {\frac {(1+x)^2}{1+x^2}} \left (1+x^2+\sqrt {1+x^2} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )\right )}{1+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 42, normalized size = 0.69
method | result | size |
default | \(\frac {\sqrt {\frac {x^{2}+2 x +1}{x^{2}+1}}\, \sqrt {x^{2}+1}\, \left (\arcsinh \left (x \right )+\sqrt {x^{2}+1}\right )}{1+x}\) | \(42\) |
risch | \(\frac {\left (x^{2}+1\right ) \sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}}{1+x}+\frac {\arcsinh \left (x \right ) \sqrt {x^{2}+1}\, \sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}}{1+x}\) | \(58\) |
trager | \(\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, \left (x^{2}+1\right )}{1+x}-\ln \left (-\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-x^{2}+\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}-x}{1+x}\right )\) | \(99\) |
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 = 0.35, size = 75, normalized size = 1.23 \begin {gather*} -\frac {{\left (x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}}}{x + 1} \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 {\frac {2 x}{x^{2} + 1} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.35, size = 49, normalized size = 0.80 \begin {gather*} -{\left (\sqrt {2} - \log \left (\sqrt {2} + 1\right )\right )} \mathrm {sgn}\left (x + 1\right ) - \log \left (-x + \sqrt {x^{2} + 1}\right ) \mathrm {sgn}\left (x + 1\right ) + \sqrt {x^{2} + 1} \mathrm {sgn}\left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\frac {2\,x}{x^2+1}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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