Optimal. Leaf size=31 \[ 2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0080735, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 63, 203} \[ 2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+x}}{1+x} \, dx &=2 \sqrt{-1+x}-2 \int \frac{1}{\sqrt{-1+x} (1+x)} \, dx\\ &=2 \sqrt{-1+x}-4 \operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\sqrt{-1+x}\right )\\ &=2 \sqrt{-1+x}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{-1+x}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0068531, size = 31, normalized size = 1. \[ 2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.8 \begin{align*} -2\,\arctan \left ( 1/2\,\sqrt{-1+x}\sqrt{2} \right ) \sqrt{2}+2\,\sqrt{-1+x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40636, size = 32, normalized size = 1.03 \begin{align*} -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87988, size = 81, normalized size = 2.61 \begin{align*} -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21423, size = 76, normalized size = 2.45 \begin{align*} \begin{cases} 2 \sqrt{x - 1} + 2 \sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 i \sqrt{1 - x} + \sqrt{2} i \log{\left (x + 1 \right )} - 2 \sqrt{2} i \log{\left (\sqrt{\frac{1}{2} - \frac{x}{2}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04958, size = 32, normalized size = 1.03 \begin{align*} -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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