3.61 \(\int \frac{(1+x)^2}{x^4 \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sqrt{1-x^2}}{3 x^3} \]

[Out]

-Sqrt[1 - x^2]/(3*x^3) - Sqrt[1 - x^2]/x^2 - (5*Sqrt[1 - x^2])/(3*x) - ArcTanh[S
qrt[1 - x^2]]

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Rubi [A]  time = 0.167301, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sqrt{1-x^2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^2/(x^4*Sqrt[1 - x^2]),x]

[Out]

-Sqrt[1 - x^2]/(3*x^3) - Sqrt[1 - x^2]/x^2 - (5*Sqrt[1 - x^2])/(3*x) - ArcTanh[S
qrt[1 - x^2]]

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Rubi in Sympy [A]  time = 12.5477, size = 49, normalized size = 0.73 \[ - \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} - \frac{5 \sqrt{- x^{2} + 1}}{3 x} - \frac{\sqrt{- x^{2} + 1}}{x^{2}} - \frac{\sqrt{- x^{2} + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**2/x**4/(-x**2+1)**(1/2),x)

[Out]

-atanh(sqrt(-x**2 + 1)) - 5*sqrt(-x**2 + 1)/(3*x) - sqrt(-x**2 + 1)/x**2 - sqrt(
-x**2 + 1)/(3*x**3)

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Mathematica [A]  time = 0.0498194, size = 47, normalized size = 0.7 \[ -\log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \left (5 x^2+3 x+1\right )}{3 x^3}+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)^2/(x^4*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]*(1 + 3*x + 5*x^2))/(3*x^3) + Log[x] - Log[1 + Sqrt[1 - x^2]]

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Maple [A]  time = 0.012, size = 56, normalized size = 0.8 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{5}{3\,x}\sqrt{-{x}^{2}+1}}-{\frac{1}{{x}^{2}}\sqrt{-{x}^{2}+1}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^2/x^4/(-x^2+1)^(1/2),x)

[Out]

-1/3*(-x^2+1)^(1/2)/x^3-5/3*(-x^2+1)^(1/2)/x-(-x^2+1)^(1/2)/x^2-arctanh(1/(-x^2+
1)^(1/2))

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Maxima [A]  time = 0.792782, size = 92, normalized size = 1.37 \[ -\frac{5 \, \sqrt{-x^{2} + 1}}{3 \, x} - \frac{\sqrt{-x^{2} + 1}}{x^{2}} - \frac{\sqrt{-x^{2} + 1}}{3 \, x^{3}} - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^4),x, algorithm="maxima")

[Out]

-5/3*sqrt(-x^2 + 1)/x - sqrt(-x^2 + 1)/x^2 - 1/3*sqrt(-x^2 + 1)/x^3 - log(2*sqrt
(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.272534, size = 194, normalized size = 2.9 \[ -\frac{5 \, x^{6} + 3 \, x^{5} - 24 \, x^{4} - 15 \, x^{3} + 15 \, x^{2} - 3 \,{\left (3 \, x^{5} - 4 \, x^{3} -{\left (x^{5} - 4 \, x^{3}\right )} \sqrt{-x^{2} + 1}\right )} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) +{\left (15 \, x^{4} + 9 \, x^{3} - 17 \, x^{2} - 12 \, x - 4\right )} \sqrt{-x^{2} + 1} + 12 \, x + 4}{3 \,{\left (3 \, x^{5} - 4 \, x^{3} -{\left (x^{5} - 4 \, x^{3}\right )} \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^4),x, algorithm="fricas")

[Out]

-1/3*(5*x^6 + 3*x^5 - 24*x^4 - 15*x^3 + 15*x^2 - 3*(3*x^5 - 4*x^3 - (x^5 - 4*x^3
)*sqrt(-x^2 + 1))*log((sqrt(-x^2 + 1) - 1)/x) + (15*x^4 + 9*x^3 - 17*x^2 - 12*x
- 4)*sqrt(-x^2 + 1) + 12*x + 4)/(3*x^5 - 4*x^3 - (x^5 - 4*x^3)*sqrt(-x^2 + 1))

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Sympy [A]  time = 18.2272, size = 128, normalized size = 1.91 \[ \begin{cases} - \frac{\sqrt{- x^{2} + 1}}{x} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3 x^{3}} & \text{for}\: x > -1 \wedge x < 1 \end{cases} + \begin{cases} - \frac{i \sqrt{x^{2} - 1}}{x} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{\sqrt{- x^{2} + 1}}{x} & \text{otherwise} \end{cases} + 2 \left (\begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x} \right )}}{2} - \frac{\sqrt{-1 + \frac{1}{x^{2}}}}{2 x} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x} \right )}}{2} - \frac{i}{2 x \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{2 x^{3} \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**2/x**4/(-x**2+1)**(1/2),x)

[Out]

Piecewise((-sqrt(-x**2 + 1)/x - (-x**2 + 1)**(3/2)/(3*x**3), (x > -1) & (x < 1))
) + Piecewise((-I*sqrt(x**2 - 1)/x, Abs(x**2) > 1), (-sqrt(-x**2 + 1)/x, True))
+ 2*Piecewise((-acosh(1/x)/2 - sqrt(-1 + x**(-2))/(2*x), Abs(x**(-2)) > 1), (I*a
sin(1/x)/2 - I/(2*x*sqrt(1 - 1/x**2)) + I/(2*x**3*sqrt(1 - 1/x**2)), True))

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GIAC/XCAS [A]  time = 0.295301, size = 169, normalized size = 2.52 \[ -\frac{x^{3}{\left (\frac{6 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{21 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{24 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}} - \frac{7 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{8 \, x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{24 \, x^{3}} +{\rm ln}\left (-\frac{\sqrt{-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^4),x, algorithm="giac")

[Out]

-1/24*x^3*(6*(sqrt(-x^2 + 1) - 1)/x - 21*(sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-
x^2 + 1) - 1)^3 - 7/8*(sqrt(-x^2 + 1) - 1)/x + 1/4*(sqrt(-x^2 + 1) - 1)^2/x^2 -
1/24*(sqrt(-x^2 + 1) - 1)^3/x^3 + ln(-(sqrt(-x^2 + 1) - 1)/abs(x))