Optimal. Leaf size=46 \[ \frac {\sqrt {x^2+1}}{x}+\sqrt {x^2+1}-\log \left (\sqrt {x^2+1}+1\right )-\frac {1}{x}-\sinh ^{-1}(x) \]
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Rubi [A] time = 0.09, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6742, 277, 215, 1591, 190, 43} \[ \frac {\sqrt {x^2+1}}{x}+\sqrt {x^2+1}-\log \left (\sqrt {x^2+1}+1\right )-\frac {1}{x}-\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 215
Rule 277
Rule 1591
Rule 6742
Rubi steps
\begin {align*} \int \frac {-1+x}{1+\sqrt {1+x^2}} \, dx &=\int \left (-\frac {1}{1+\sqrt {1+x^2}}+\frac {x}{1+\sqrt {1+x^2}}\right ) \, dx\\ &=-\int \frac {1}{1+\sqrt {1+x^2}} \, dx+\int \frac {x}{1+\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {x}} \, dx,x,1+x^2\right )-\int \left (-\frac {1}{x^2}+\frac {\sqrt {1+x^2}}{x^2}\right ) \, dx\\ &=-\frac {1}{x}-\int \frac {\sqrt {1+x^2}}{x^2} \, dx+\operatorname {Subst}\left (\int \frac {x}{1+x} \, dx,x,\sqrt {1+x^2}\right )\\ &=-\frac {1}{x}+\frac {\sqrt {1+x^2}}{x}-\int \frac {1}{\sqrt {1+x^2}} \, dx+\operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,\sqrt {1+x^2}\right )\\ &=-\frac {1}{x}+\sqrt {1+x^2}+\frac {\sqrt {1+x^2}}{x}-\sinh ^{-1}(x)-\log \left (1+\sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.00 \[ \frac {\sqrt {x^2+1}}{x}+\sqrt {x^2+1}-\log \left (\sqrt {x^2+1}+1\right )-\frac {1}{x}-\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 64, normalized size = 1.39 \[ \frac {x \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + 1\right )} + x + 1\right ) - x \log \relax (x) - x \log \left (-x + \sqrt {x^{2} + 1} + 1\right ) + \sqrt {x^{2} + 1} {\left (x + 1\right )} + x - 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 79, normalized size = 1.72 \[ \sqrt {x^{2} + 1} - \frac {2}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} - \frac {1}{x} + \log \left (-x + \sqrt {x^{2} + 1}\right ) - \log \left ({\left | x \right |}\right ) - \log \left ({\left | -x + \sqrt {x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 53, normalized size = 1.15 \[ -\sqrt {x^{2}+1}\, x -\arcsinh \relax (x )-\arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right )-\ln \relax (x )-\frac {1}{x}+\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{x}+\sqrt {x^{2}+1} \]
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 \[ \frac {1}{4} \, x^{2} - \frac {1}{2} \, x - \int \frac {x^{3} - x^{2}}{2 \, {\left (x^{2} + 2 \, \sqrt {x^{2} + 1} + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 46, normalized size = 1.00 \[ \sqrt {x^2+1}-\ln \relax (x)-\mathrm {asinh}\relax (x)+\frac {\sqrt {x^2+1}}{x}-\frac {1}{x}+\mathrm {atan}\left (\sqrt {x^2+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.98, size = 48, normalized size = 1.04 \[ \frac {x}{\sqrt {x^{2} + 1}} + \sqrt {x^{2} + 1} - \log {\left (\sqrt {x^{2} + 1} + 1 \right )} - \operatorname {asinh}{\relax (x )} - \frac {1}{x} + \frac {1}{x \sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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