Optimal. Leaf size=109 \[ \frac {x+1}{\sqrt {\frac {2 x}{x^2+1}+1}}-\frac {(x+1) \sinh ^{-1}(x)}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {\sqrt {2} (x+1) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}} \]
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Rubi [A] time = 0.06, 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 = {6723, 970, 735, 844, 215, 725, 206} \[ \frac {x+1}{\sqrt {\frac {2 x}{x^2+1}+1}}-\frac {(x+1) \sinh ^{-1}(x)}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {\sqrt {2} (x+1) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 725
Rule 735
Rule 844
Rule 970
Rule 6723
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)) \operatorname {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 = 0.03, size = 72, normalized size = 0.66 \[ \frac {(x+1) \left (\sqrt {x^2+1}-\sqrt {2} \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )-\sinh ^{-1}(x)\right )}{\sqrt {\frac {(x+1)^2}{x^2+1}} \sqrt {x^2+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 142, normalized size = 1.30 \[ \frac {\sqrt {2} {\left (x + 1\right )} \log \left (-\frac {x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} - 1}{x^{2} + 2 \, x + 1}\right ) + {\left (x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) + {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 88, normalized size = 0.81 \[ \frac {\sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\log \left (-x + \sqrt {x^{2} + 1}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\sqrt {x^{2} + 1}}{\mathrm {sgn}\left (x + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 0.72 \[ \frac {x +1}{\sqrt {\frac {\left (x +1\right )^{2}}{x^{2}+1}}}+\frac {\left (-\arcsinh \relax (x )-\sqrt {2}\, \arctanh \left (\frac {\left (-2 x +2\right ) \sqrt {2}}{4 \sqrt {-2 x +\left (x +1\right )^{2}}}\right )\right ) \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{x^{2}+1}}\, \sqrt {x^{2}+1}} \]
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 \[ \int \frac {1}{\sqrt {\frac {2 \, x}{x^{2} + 1} + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {2\,x}{x^2+1}+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\frac {2 x}{x^{2} + 1} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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