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 = 0.0857062, antiderivative size = 46, normalized size of antiderivative = 1., 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 6742
Rule 277
Rule 215
Rule 1591
Rule 190
Rule 43
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*}
Mathematica [A] time = 0.0348584, size = 46, normalized size = 1. \[ \frac{\sqrt{x^2+1}}{x}+\sqrt{x^2+1}-\log \left (\sqrt{x^2+1}+1\right )-\frac{1}{x}-\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 53, normalized size = 1.2 \begin{align*} -{x}^{-1}+\sqrt{{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) -\ln \left ( x \right ) +{\frac{1}{x} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-x\sqrt{{x}^{2}+1}-{\it Arcsinh} \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71427, size = 171, normalized size = 3.72 \begin{align*} \frac{x \log \left (2 \, x^{2} - \sqrt{x^{2} + 1}{\left (2 \, x + 1\right )} + x + 1\right ) - x \log \left (x\right ) - x \log \left (-x + \sqrt{x^{2} + 1} + 1\right ) + \sqrt{x^{2} + 1}{\left (x + 1\right )} + x - 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.22857, size = 48, normalized size = 1.04 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} - \log{\left (\sqrt{x^{2} + 1} + 1 \right )} - \operatorname{asinh}{\left (x \right )} - \frac{1}{x} + \frac{1}{x \sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09697, size = 107, normalized size = 2.33 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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