Optimal. Leaf size=33 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]
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Rubi [A] time = 0.142982, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2104, 6742, 38, 52} \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 2104
Rule 6742
Rule 38
Rule 52
Rubi steps
\begin{align*} \int \frac{-\sqrt{-1+x}+\sqrt{1+x}}{\sqrt{-1+x}+\sqrt{1+x}} \, dx &=-\left (\frac{1}{2} \int \sqrt{-1+x} \left (-\sqrt{-1+x}+\sqrt{1+x}\right ) \, dx\right )+\frac{1}{2} \int \sqrt{1+x} \left (-\sqrt{-1+x}+\sqrt{1+x}\right ) \, dx\\ &=\frac{1}{2} \int \left (1+x-\sqrt{-1+x} \sqrt{1+x}\right ) \, dx-\frac{1}{2} \int \left (1-x+\sqrt{-1+x} \sqrt{1+x}\right ) \, dx\\ &=\frac{x^2}{2}-2 \left (\frac{1}{2} \int \sqrt{-1+x} \sqrt{1+x} \, dx\right )\\ &=\frac{x^2}{2}-2 \left (\frac{1}{4} \sqrt{-1+x} x \sqrt{1+x}-\frac{1}{4} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx\right )\\ &=\frac{x^2}{2}-2 \left (\frac{1}{4} \sqrt{-1+x} x \sqrt{1+x}-\frac{1}{4} \cosh ^{-1}(x)\right )\\ \end{align*}
Mathematica [A] time = 0.161114, size = 58, normalized size = 1.76 \[ \frac{1}{2} \left (x^2-\sqrt{x-1} \sqrt{x+1} x+\frac{2 (x-1) \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )}{\sqrt{-(x-1)^2}}+1\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.004, size = 62, normalized size = 1.9 \begin{align*} -{\frac{1}{2}\sqrt{x-1} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{x-1}\sqrt{1+x}}+{\frac{1}{2}\sqrt{ \left ( x-1 \right ) \left ( 1+x \right ) }\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{x-1}}}{\frac{1}{\sqrt{1+x}}}}+{\frac{{x}^{2}}{2}} \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*} \int \frac{\sqrt{x + 1} - \sqrt{x - 1}}{\sqrt{x + 1} + \sqrt{x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.958409, size = 109, normalized size = 3.3 \begin{align*} -\frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x + \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.2528, size = 226, normalized size = 6.85 \begin{align*} - \frac{\left (x - 1\right )^{\frac{5}{2}}}{4 \sqrt{x + 1}} - \frac{3 \left (x - 1\right )^{\frac{3}{2}}}{4 \sqrt{x + 1}} - \frac{\sqrt{x - 1}}{2 \sqrt{x + 1}} + \frac{\left (x - 1\right )^{2}}{4} + 2 \left (\begin{cases} \frac{\left (x + 1\right )^{2}}{8} + \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{8 \sqrt{x - 1}} + \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{8 \sqrt{x - 1}} - \frac{\sqrt{x + 1}}{4 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{\left (x + 1\right )^{2}}{8} - \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{8 \sqrt{1 - x}} - \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{8 \sqrt{1 - x}} + \frac{i \sqrt{x + 1}}{4 \sqrt{1 - x}} & \text{otherwise} \end{cases}\right ) + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1937, size = 57, normalized size = 1.73 \begin{align*} \frac{1}{2} \,{\left (x + 1\right )}^{2} - \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x - x - \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) - 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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