3.514 \(\int \frac{\sqrt{-1+\frac{1}{x^2}} \left (-1+x^2\right )^3}{x} \, dx\)

Optimal. Leaf size=76 \[ -\frac{35}{48} \left (\frac{1}{x^2}-1\right )^{3/2} x^2+\frac{35}{16} \sqrt{\frac{1}{x^2}-1}-\frac{35}{16} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right )-\frac{1}{6} \left (\frac{1}{x^2}-1\right )^{7/2} x^6-\frac{7}{24} \left (\frac{1}{x^2}-1\right )^{5/2} x^4 \]

[Out]

(35*Sqrt[-1 + x^(-2)])/16 - (35*(-1 + x^(-2))^(3/2)*x^2)/48 - (7*(-1 + x^(-2))^(
5/2)*x^4)/24 - ((-1 + x^(-2))^(7/2)*x^6)/6 - (35*ArcTan[Sqrt[-1 + x^(-2)]])/16

_______________________________________________________________________________________

Rubi [A]  time = 0.0693928, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{35}{48} \left (\frac{1}{x^2}-1\right )^{3/2} x^2+\frac{35}{16} \sqrt{\frac{1}{x^2}-1}-\frac{35}{16} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right )-\frac{1}{6} \left (\frac{1}{x^2}-1\right )^{7/2} x^6-\frac{7}{24} \left (\frac{1}{x^2}-1\right )^{5/2} x^4 \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + x^(-2)]*(-1 + x^2)^3)/x,x]

[Out]

(35*Sqrt[-1 + x^(-2)])/16 - (35*(-1 + x^(-2))^(3/2)*x^2)/48 - (7*(-1 + x^(-2))^(
5/2)*x^4)/24 - ((-1 + x^(-2))^(7/2)*x^6)/6 - (35*ArcTan[Sqrt[-1 + x^(-2)]])/16

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.47909, size = 76, normalized size = 1. \[ - \frac{x^{6} \left (-1 + \frac{1}{x^{2}}\right )^{\frac{7}{2}}}{6} - \frac{7 x^{4} \left (-1 + \frac{1}{x^{2}}\right )^{\frac{5}{2}}}{24} - \frac{35 x^{2} \left (-1 + \frac{1}{x^{2}}\right )^{\frac{3}{2}}}{48} + \frac{35 \sqrt{-1 + \frac{1}{x^{2}}}}{16} - \frac{35 \operatorname{atan}{\left (\sqrt{-1 + \frac{1}{x^{2}}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-1)**3*(-1+1/x**2)**(1/2)/x,x)

[Out]

-x**6*(-1 + x**(-2))**(7/2)/6 - 7*x**4*(-1 + x**(-2))**(5/2)/24 - 35*x**2*(-1 +
x**(-2))**(3/2)/48 + 35*sqrt(-1 + x**(-2))/16 - 35*atan(sqrt(-1 + x**(-2)))/16

_______________________________________________________________________________________

Mathematica [A]  time = 0.0451784, size = 65, normalized size = 0.86 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (\sqrt{x^2-1} \left (8 x^6-38 x^4+87 x^2+48\right )-105 x \log \left (\sqrt{x^2-1}+x\right )\right )}{48 \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + x^(-2)]*(-1 + x^2)^3)/x,x]

[Out]

(Sqrt[-1 + x^(-2)]*(Sqrt[-1 + x^2]*(48 + 87*x^2 - 38*x^4 + 8*x^6) - 105*x*Log[x
+ Sqrt[-1 + x^2]]))/(48*Sqrt[-1 + x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 83, normalized size = 1.1 \[{\frac{1}{48}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}} \left ( -8\,{x}^{4} \left ( -{x}^{2}+1 \right ) ^{3/2}+30\,{x}^{2} \left ( -{x}^{2}+1 \right ) ^{3/2}+48\, \left ( -{x}^{2}+1 \right ) ^{3/2}+105\,{x}^{2}\sqrt{-{x}^{2}+1}+105\,\arcsin \left ( x \right ) x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-1)^3*(-1+1/x^2)^(1/2)/x,x)

[Out]

1/48*(-(x^2-1)/x^2)^(1/2)*(-8*x^4*(-x^2+1)^(3/2)+30*x^2*(-x^2+1)^(3/2)+48*(-x^2+
1)^(3/2)+105*x^2*(-x^2+1)^(1/2)+105*arcsin(x)*x)/(-x^2+1)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.804317, size = 162, normalized size = 2.13 \[ \frac{3}{2} \, x^{2} \sqrt{\frac{1}{x^{2}} - 1} + \sqrt{\frac{1}{x^{2}} - 1} - \frac{3 \,{\left (\frac{1}{x^{2}} - 1\right )}^{\frac{5}{2}} + 8 \,{\left (\frac{1}{x^{2}} - 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{\frac{1}{x^{2}} - 1}}{48 \,{\left ({\left (\frac{1}{x^{2}} - 1\right )}^{3} + 3 \,{\left (\frac{1}{x^{2}} - 1\right )}^{2} + \frac{3}{x^{2}} - 2\right )}} + \frac{3 \,{\left ({\left (\frac{1}{x^{2}} - 1\right )}^{\frac{3}{2}} - \sqrt{\frac{1}{x^{2}} - 1}\right )}}{8 \,{\left ({\left (\frac{1}{x^{2}} - 1\right )}^{2} + \frac{2}{x^{2}} - 1\right )}} - \frac{35}{16} \, \arctan \left (\sqrt{\frac{1}{x^{2}} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^3*sqrt(1/x^2 - 1)/x,x, algorithm="maxima")

[Out]

3/2*x^2*sqrt(1/x^2 - 1) + sqrt(1/x^2 - 1) - 1/48*(3*(1/x^2 - 1)^(5/2) + 8*(1/x^2
- 1)^(3/2) - 3*sqrt(1/x^2 - 1))/((1/x^2 - 1)^3 + 3*(1/x^2 - 1)^2 + 3/x^2 - 2) +
3/8*((1/x^2 - 1)^(3/2) - sqrt(1/x^2 - 1))/((1/x^2 - 1)^2 + 2/x^2 - 1) - 35/16*a
rctan(sqrt(1/x^2 - 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.268988, size = 74, normalized size = 0.97 \[ \frac{1}{48} \,{\left (8 \, x^{6} - 38 \, x^{4} + 87 \, x^{2} + 48\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}} - \frac{35}{8} \, \arctan \left (\frac{x \sqrt{-\frac{x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^3*sqrt(1/x^2 - 1)/x,x, algorithm="fricas")

[Out]

1/48*(8*x^6 - 38*x^4 + 87*x^2 + 48)*sqrt(-(x^2 - 1)/x^2) - 35/8*arctan((x*sqrt(-
(x^2 - 1)/x^2) - 1)/x)

_______________________________________________________________________________________

Sympy [A]  time = 32.4403, size = 348, normalized size = 4.58 \[ - \begin{cases} - \frac{i x}{\sqrt{x^{2} - 1}} + i \operatorname{acosh}{\left (x \right )} + \frac{i}{x \sqrt{x^{2} - 1}} & \text{for}\: \left |{x^{2}}\right | > 1 \\\frac{x}{\sqrt{- x^{2} + 1}} - \operatorname{asin}{\left (x \right )} - \frac{1}{x \sqrt{- x^{2} + 1}} & \text{otherwise} \end{cases} + 3 \left (\begin{cases} \frac{i x^{3}}{2 \sqrt{x^{2} - 1}} - \frac{i x}{2 \sqrt{x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (x \right )}}{2} & \text{for}\: \left |{x^{2}}\right | > 1 \\\frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} & \text{otherwise} \end{cases}\right ) - 3 \left (\begin{cases} \frac{i x^{5}}{4 \sqrt{x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{x^{2} - 1}} + \frac{i x}{8 \sqrt{x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (x \right )}}{8} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{x^{5}}{4 \sqrt{- x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- x^{2} + 1}} - \frac{x}{8 \sqrt{- x^{2} + 1}} + \frac{\operatorname{asin}{\left (x \right )}}{8} & \text{otherwise} \end{cases}\right ) + \begin{cases} \frac{i x^{7}}{6 \sqrt{x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{x^{2} - 1}} - \frac{i x^{3}}{48 \sqrt{x^{2} - 1}} + \frac{i x}{16 \sqrt{x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (x \right )}}{16} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{x^{7}}{6 \sqrt{- x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- x^{2} + 1}} + \frac{x^{3}}{48 \sqrt{- x^{2} + 1}} - \frac{x}{16 \sqrt{- x^{2} + 1}} + \frac{\operatorname{asin}{\left (x \right )}}{16} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-1)**3*(-1+1/x**2)**(1/2)/x,x)

[Out]

-Piecewise((-I*x/sqrt(x**2 - 1) + I*acosh(x) + I/(x*sqrt(x**2 - 1)), Abs(x**2) >
1), (x/sqrt(-x**2 + 1) - asin(x) - 1/(x*sqrt(-x**2 + 1)), True)) + 3*Piecewise(
(I*x**3/(2*sqrt(x**2 - 1)) - I*x/(2*sqrt(x**2 - 1)) - I*acosh(x)/2, Abs(x**2) >
1), (x*sqrt(-x**2 + 1)/2 + asin(x)/2, True)) - 3*Piecewise((I*x**5/(4*sqrt(x**2
- 1)) - 3*I*x**3/(8*sqrt(x**2 - 1)) + I*x/(8*sqrt(x**2 - 1)) - I*acosh(x)/8, Abs
(x**2) > 1), (-x**5/(4*sqrt(-x**2 + 1)) + 3*x**3/(8*sqrt(-x**2 + 1)) - x/(8*sqrt
(-x**2 + 1)) + asin(x)/8, True)) + Piecewise((I*x**7/(6*sqrt(x**2 - 1)) - 5*I*x*
*5/(24*sqrt(x**2 - 1)) - I*x**3/(48*sqrt(x**2 - 1)) + I*x/(16*sqrt(x**2 - 1)) -
I*acosh(x)/16, Abs(x**2) > 1), (-x**7/(6*sqrt(-x**2 + 1)) + 5*x**5/(24*sqrt(-x**
2 + 1)) + x**3/(48*sqrt(-x**2 + 1)) - x/(16*sqrt(-x**2 + 1)) + asin(x)/16, True)
)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285528, size = 104, normalized size = 1.37 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, x^{2}{\rm sign}\left (x\right ) - 19 \,{\rm sign}\left (x\right )\right )} x^{2} + 87 \,{\rm sign}\left (x\right )\right )} \sqrt{-x^{2} + 1} x + \frac{35}{16} \, \arcsin \left (x\right ){\rm sign}\left (x\right ) - \frac{x{\rm sign}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}{\rm sign}\left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^3*sqrt(1/x^2 - 1)/x,x, algorithm="giac")

[Out]

1/48*(2*(4*x^2*sign(x) - 19*sign(x))*x^2 + 87*sign(x))*sqrt(-x^2 + 1)*x + 35/16*
arcsin(x)*sign(x) - 1/2*x*sign(x)/(sqrt(-x^2 + 1) - 1) + 1/2*(sqrt(-x^2 + 1) - 1
)*sign(x)/x