3.1412 \(\int \frac{\sqrt{-1+x}}{(1+x)^3} \, dx\)

Optimal. Leaf size=56 \[ \frac{\sqrt{x-1}}{8 (x+1)}-\frac{\sqrt{x-1}}{2 (x+1)^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

[Out]

-Sqrt[-1 + x]/(2*(1 + x)^2) + Sqrt[-1 + x]/(8*(1 + x)) + ArcTan[Sqrt[-1 + x]/Sqr
t[2]]/(8*Sqrt[2])

_______________________________________________________________________________________

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]

-Sqrt[-1 + x]/(2*(1 + x)^2) + Sqrt[-1 + x]/(8*(1 + x)) + ArcTan[Sqrt[-1 + x]/Sqr
t[2]]/(8*Sqrt[2])

_______________________________________________________________________________________

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]

sqrt(x - 1)/(8*(x + 1)) - sqrt(x - 1)/(2*(x + 1)**2) + sqrt(2)*atan(sqrt(2)*sqrt
(x - 1)/2)/16

_______________________________________________________________________________________

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]

((2*(-3 + x)*Sqrt[-1 + x])/(1 + x)^2 + Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]])/16

_______________________________________________________________________________________

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]

2*(1/16*(-1+x)^(3/2)-1/8*(-1+x)^(1/2))/(1+x)^2+1/16*arctan(1/2*(-1+x)^(1/2)*2^(1
/2))*2^(1/2)

_______________________________________________________________________________________

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]

1/16*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x - 1)) + 1/8*((x - 1)^(3/2) - 2*sqrt(x - 1
))/((x - 1)^2 + 4*x)

_______________________________________________________________________________________

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]

1/16*sqrt(2)*(sqrt(2)*sqrt(x - 1)*(x - 3) + (x^2 + 2*x + 1)*arctan(1/2*sqrt(2)*s
qrt(x - 1)))/(x^2 + 2*x + 1)

_______________________________________________________________________________________

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)

[Out]

Piecewise((sqrt(2)*I*acosh(sqrt(2)/sqrt(x + 1))/16 - I/(8*sqrt(-1 + 2/(x + 1))*s
qrt(x + 1)) + 3*I/(4*sqrt(-1 + 2/(x + 1))*(x + 1)**(3/2)) - I/(sqrt(-1 + 2/(x +
1))*(x + 1)**(5/2)), 2*Abs(1/(x + 1)) > 1), (-sqrt(2)*asin(sqrt(2)/sqrt(x + 1))/
16 + 1/(8*sqrt(1 - 2/(x + 1))*sqrt(x + 1)) - 3/(4*sqrt(1 - 2/(x + 1))*(x + 1)**(
3/2)) + 1/(sqrt(1 - 2/(x + 1))*(x + 1)**(5/2)), True))

_______________________________________________________________________________________

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]

1/16*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x - 1)) + 1/8*((x - 1)^(3/2) - 2*sqrt(x - 1
))/(x + 1)^2