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

Optimal. Leaf size=43 \[ \frac{4 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0848861, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{4 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.67878, size = 32, normalized size = 0.74 \[ - \operatorname{asin}{\left (x \right )} - \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} + \frac{4 \sqrt{x + 1}}{\sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0890679, size = 101, normalized size = 2.35 \[ -\frac{4 \sqrt{1-x^2}}{x-1}+\log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{1-x}-\sqrt{x+1}+2\right )-\log \left (\sqrt{x+1}+1\right )+\log \left (\sqrt{1-x}+\sqrt{x+1}+2\right )-2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.018, size = 70, normalized size = 1.6 \[{\frac{1}{-1+x} \left ( -{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x-\arcsin \left ( x \right ) x+{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) +\arcsin \left ( x \right ) -4\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.767473, size = 72, normalized size = 1.67 \[ \frac{4 \, x}{\sqrt{-x^{2} + 1}} + \frac{4}{\sqrt{-x^{2} + 1}} - \arcsin \left (x\right ) - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(x*(-x + 1)^(3/2)),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 = 0.279741, size = 132, normalized size = 3.07 \[ \frac{2 \,{\left (x + \sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) +{\left (x + \sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 8 \, x}{x + \sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError