Optimal. Leaf size=109 \[ \frac {1+x}{\sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {\sqrt {2} (1+x) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x^2}}\right )}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}} \]
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Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6855, 984, 749,
858, 221, 739, 212} \begin {gather*} \frac {x+1}{\sqrt {\frac {2 x}{x^2+1}+1}}-\frac {(x+1) \sinh ^{-1}(x)}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {\sqrt {2} (x+1) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 739
Rule 749
Rule 858
Rule 984
Rule 6855
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\frac {2 x}{1+x^2}}} \, dx &=\frac {\sqrt {1+2 x+x^2} \int \frac {\sqrt {1+x^2}}{\sqrt {1+2 x+x^2}} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {(2+2 x) \int \frac {\sqrt {1+x^2}}{2+2 x} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {1+x}{\sqrt {1+\frac {2 x}{1+x^2}}}+\frac {(2+2 x) \int \frac {2-2 x}{(2+2 x) \sqrt {1+x^2}} \, dx}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {1+x}{\sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(2+2 x) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}+\frac {(2 (2+2 x)) \int \frac {1}{(2+2 x) \sqrt {1+x^2}} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {1+x}{\sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(2 (2+2 x)) \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {2-2 x}{\sqrt {1+x^2}}\right )}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {1+x}{\sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {\sqrt {2} (1+x) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x^2}}\right )}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 82, normalized size = 0.75 \begin {gather*} \frac {(1+x) \left (\sqrt {1+x^2}-\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {1+x-\sqrt {1+x^2}}{\sqrt {2}}\right )\right )}{\sqrt {\frac {(1+x)^2}{1+x^2}} \sqrt {1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 79, normalized size = 0.72
method | result | size |
risch | \(\frac {1+x}{\sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}}+\frac {\left (-\arcsinh \left (x \right )-\sqrt {2}\, \arctanh \left (\frac {\left (2-2 x \right ) \sqrt {2}}{4 \sqrt {\left (1+x \right )^{2}-2 x}}\right )\right ) \left (1+x \right )}{\sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}\, \sqrt {x^{2}+1}}\) | \(79\) |
trager | \(\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, \left (x^{2}+1\right )}{1+x}+\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+2 \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}-2\right )+2 \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{\left (1+x \right )^{2}}\right )+\ln \left (-\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-x^{2}+\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}-x}{1+x}\right )\) | \(179\) |
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 = 142, normalized size = 1.30 \begin {gather*} \frac {\sqrt {2} {\left (x + 1\right )} \log \left (-\frac {x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} - 1}{x^{2} + 2 \, x + 1}\right ) + {\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 \frac {1}{\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 = 5.72, size = 88, normalized size = 0.81 \begin {gather*} \frac {\sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\log \left (-x + \sqrt {x^{2} + 1}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\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 = -0.01 \begin {gather*} \int \frac {1}{\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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