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

Optimal. Leaf size=35 \[ \frac{4 (x+1)}{\sqrt{1-x^2}}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\sin ^{-1}(x) \]

[Out]

(4*(1 + x))/Sqrt[1 - x^2] - ArcSin[x] - ArcTanh[Sqrt[1 - x^2]]

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Rubi [A]  time = 0.12164, antiderivative size = 35, 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 \[ \frac{4 (x+1)}{\sqrt{1-x^2}}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(4*(1 + x))/Sqrt[1 - x^2] - ArcSin[x] - ArcTanh[Sqrt[1 - x^2]]

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Rubi in Sympy [A]  time = 5.15534, size = 26, normalized size = 0.74 \[ - \operatorname{asin}{\left (x \right )} - \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} + \frac{4 \sqrt{- x^{2} + 1}}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asin(x) - atanh(sqrt(-x**2 + 1)) + 4*sqrt(-x**2 + 1)/(-x + 1)

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Mathematica [A]  time = 0.0435132, size = 41, normalized size = 1.17 \[ -\frac{4 \sqrt{1-x^2}}{x-1}-\log \left (\sqrt{1-x^2}+1\right )+\log (x)-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[1 - x^2])/(-1 + x) - ArcSin[x] + Log[x] - Log[1 + Sqrt[1 - x^2]]

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Maple [A]  time = 0.011, size = 41, normalized size = 1.2 \[ 4\,{\frac{1}{\sqrt{-{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) +4\,{\frac{x}{\sqrt{-{x}^{2}+1}}}-\arcsin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^3/x/(-x^2+1)^(3/2),x)

[Out]

4/(-x^2+1)^(1/2)-arctanh(1/(-x^2+1)^(1/2))+4*x/(-x^2+1)^(1/2)-arcsin(x)

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Maxima [A]  time = 0.758425, size = 72, normalized size = 2.06 \[ \frac{4 \, x}{\sqrt{-x^{2} + 1}} + \frac{4}{\sqrt{-x^{2} + 1}} - \arcsin \left (x\right ) - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x/sqrt(-x^2 + 1) + 4/sqrt(-x^2 + 1) - arcsin(x) - log(2*sqrt(-x^2 + 1)/abs(x)
+ 2/abs(x))

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Fricas [A]  time = 0.27733, size = 105, normalized size = 3. \[ \frac{2 \,{\left (x + \sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) +{\left (x + \sqrt{-x^{2} + 1} - 1\right )} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + 8 \, x}{x + \sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + 1\right )^{3}}{x \left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 1)**3/(x*(-(x - 1)*(x + 1))**(3/2)), x)

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GIAC/XCAS [A]  time = 0.273532, size = 59, normalized size = 1.69 \[ \frac{8}{\frac{\sqrt{-x^{2} + 1} - 1}{x} + 1} - \arcsin \left (x\right ) +{\rm ln}\left (-\frac{\sqrt{-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/((sqrt(-x^2 + 1) - 1)/x + 1) - arcsin(x) + ln(-(sqrt(-x^2 + 1) - 1)/abs(x))