Optimal. Leaf size=28 \[ -\frac {(1-x) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x} \]
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Rubi [A]
time = 0.08, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6855, 984, 651}
\begin {gather*} -\frac {(1-x) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 984
Rule 6855
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\frac {2 x}{1+x^2}}}{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}}{\left (1+x^2\right )^{3/2}} \, 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}{\left (1+x^2\right )^{3/2}} \, dx}{2+2 x}\\ &=-\frac {(1-x) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 25, normalized size = 0.89 \begin {gather*} \frac {(-1+x) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 28, normalized size = 1.00
method | result | size |
risch | \(\frac {\sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}\, \left (-1+x \right )}{1+x}\) | \(25\) |
gosper | \(\frac {\left (-1+x \right ) \sqrt {\frac {x^{2}+2 x +1}{x^{2}+1}}}{1+x}\) | \(28\) |
default | \(\frac {\left (-1+x \right ) \sqrt {\frac {x^{2}+2 x +1}{x^{2}+1}}}{1+x}\) | \(28\) |
trager | \(\frac {\left (-1+x \right ) \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{1+x}\) | \(31\) |
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 = 0.38, size = 31, normalized size = 1.11 \begin {gather*} \frac {{\left (x - 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x + 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 \frac {\sqrt {\frac {\left (x + 1\right )^{2}}{x^{2} + 1}}}{x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.22, size = 30, normalized size = 1.07 \begin {gather*} \sqrt {2} \mathrm {sgn}\left (x + 1\right ) + \frac {x \mathrm {sgn}\left (x + 1\right ) - \mathrm {sgn}\left (x + 1\right )}{\sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.52, size = 23, normalized size = 0.82 \begin {gather*} \frac {\sqrt {\frac {2\,x}{x^2+1}+1}\,\left (x-1\right )}{x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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