Optimal. Leaf size=28 \[ -\frac {(1-x) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1} \]
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Rubi [A] time = 0.12, 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 = {6723, 970, 637} \[ -\frac {(1-x) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1} \]
Antiderivative was successfully verified.
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Rule 637
Rule 970
Rule 6723
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 = 0.03, size = 26, normalized size = 0.93 \[ \frac {(x-1) \sqrt {\frac {(x+1)^2}{x^2+1}}}{x+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 31, normalized size = 1.11 \[ \frac {{\left (x - 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x + 1}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 30, normalized size = 1.07 \[ \sqrt {2} \mathrm {sgn}\left (x + 1\right ) + \frac {x \mathrm {sgn}\left (x + 1\right ) - \mathrm {sgn}\left (x + 1\right )}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {\left (x -1\right ) \sqrt {\frac {x^{2}+2 x +1}{x^{2}+1}}}{x +1} \]
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 \[ \int \frac {\sqrt {\frac {2 \, x}{x^{2} + 1} + 1}}{x^{2} + 1}\,{d x} \]
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 \[ \frac {\sqrt {\frac {2\,x}{x^2+1}+1}\,\left (x-1\right )}{x+1} \]
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 \[ \int \frac {\sqrt {\frac {\left (x + 1\right )^{2}}{x^{2} + 1}}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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