Optimal. Leaf size=56 \[ \frac{\sqrt{x-1}}{8 (x+1)}-\frac{\sqrt{x-1}}{2 (x+1)^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0400785, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x-1}}{8 (x+1)}-\frac{\sqrt{x-1}}{2 (x+1)^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-1 + x]/(1 + x)^3,x]
[Out]
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Rubi in Sympy [A] time = 5.44703, size = 46, normalized size = 0.82 \[ \frac{\sqrt{x - 1}}{8 \left (x + 1\right )} - \frac{\sqrt{x - 1}}{2 \left (x + 1\right )^{2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+x)**(1/2)/(1+x)**3,x)
[Out]
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Mathematica [A] time = 0.0373926, size = 42, normalized size = 0.75 \[ \frac{1}{16} \left (\frac{2 \sqrt{x-1} (x-3)}{(x+1)^2}+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-1 + x]/(1 + x)^3,x]
[Out]
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Maple [A] time = 0.011, size = 40, normalized size = 0.7 \[ 2\,{\frac{1/16\, \left ( -1+x \right ) ^{3/2}-1/8\,\sqrt{-1+x}}{ \left ( 1+x \right ) ^{2}}}+{\frac{\sqrt{2}}{16}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1+x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+x)^(1/2)/(1+x)^3,x)
[Out]
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Maxima [A] time = 1.52248, size = 58, normalized size = 1.04 \[ \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + \frac{{\left (x - 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{x - 1}}{8 \,{\left ({\left (x - 1\right )}^{2} + 4 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(x + 1)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216842, size = 65, normalized size = 1.16 \[ \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{x - 1}{\left (x - 3\right )} +{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right )\right )}}{16 \,{\left (x^{2} + 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(x + 1)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.04309, size = 168, normalized size = 3. \[ \begin{cases} \frac{\sqrt{2} i \operatorname{acosh}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )}}{16} - \frac{i}{8 \sqrt{-1 + \frac{2}{x + 1}} \sqrt{x + 1}} + \frac{3 i}{4 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{\frac{3}{2}}} - \frac{i}{\sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{\frac{5}{2}}} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{\sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )}}{16} + \frac{1}{8 \sqrt{1 - \frac{2}{x + 1}} \sqrt{x + 1}} - \frac{3}{4 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{\frac{3}{2}}} + \frac{1}{\sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+x)**(1/2)/(1+x)**3,x)
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GIAC/XCAS [A] time = 0.216032, size = 50, normalized size = 0.89 \[ \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + \frac{{\left (x - 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{x - 1}}{8 \,{\left (x + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(x + 1)^3,x, algorithm="giac")
[Out]