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

Optimal. Leaf size=68 \[ \frac{\sqrt{(x-1)^3} \tan ^{-1}\left (\sqrt{x-1}\right )}{(x-1)^{3/2}}+\tan ^{-1}\left (\sqrt{x-1}\right )-\frac{\sqrt{(x-1)^3} \tanh ^{-1}\left (\sqrt{x-1}\right )}{(x-1)^{3/2}}+\tanh ^{-1}\left (\sqrt{x-1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.255928, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{\sqrt{(x-1)^3} \tan ^{-1}\left (\sqrt{x-1}\right )}{(x-1)^{3/2}}+\tan ^{-1}\left (\sqrt{x-1}\right )-\frac{\sqrt{(x-1)^3} \tanh ^{-1}\left (\sqrt{x-1}\right )}{(x-1)^{3/2}}+\tanh ^{-1}\left (\sqrt{x-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \sqrt{x - 1} + \frac{\sqrt{\left (x - 1\right )^{3}}}{3} + \operatorname{atan}{\left (\sqrt{x - 1} \right )} + 2 \int ^{\sqrt{x - 1}} \frac{x - \sqrt{x^{6}}}{x}\, dx + 2 \int ^{\sqrt{x - 1}} \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x - 1}\, dx + 2 \int ^{\sqrt{x - 1}} \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x + 1}\, dx + \frac{\sqrt{\left (x - 1\right )^{3}} \operatorname{atan}{\left (\sqrt{x - 1} \right )}}{\left (x - 1\right )^{\frac{3}{2}}} + \frac{\sqrt{\left (x - 1\right )^{3}}}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x - 1) + sqrt((x - 1)**3)/3 + atan(sqrt(x - 1)) + 2*Integral((x - sqrt(x**
6))/x, (x, sqrt(x - 1))) + 2*Integral((-x/4 + sqrt(x**6)/4)/(x - 1), (x, sqrt(x
- 1))) + 2*Integral((-x/4 + sqrt(x**6)/4)/(x + 1), (x, sqrt(x - 1))) + sqrt((x -
1)**3)*atan(sqrt(x - 1))/(x - 1)**(3/2) + sqrt((x - 1)**3)/(-x + 1)

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Mathematica [A]  time = 0.0381708, size = 51, normalized size = 0.75 \[ \tan ^{-1}\left (\sqrt{x-1}\right )+\tan ^{-1}\left (\frac{\sqrt{(x-1)^3}}{x-1}\right )+\tanh ^{-1}\left (\sqrt{x-1}\right )-\tanh ^{-1}\left (\frac{\sqrt{(x-1)^3}}{x-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[-1 + x]] + ArcTan[Sqrt[(-1 + x)^3]/(-1 + x)] + ArcTanh[Sqrt[-1 + x]]
- ArcTanh[Sqrt[(-1 + x)^3]/(-1 + x)]

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Maple [A]  time = 0.018, size = 40, normalized size = 0.6 \[ 2\,{1\arctan \left ( \sqrt{{\frac{\sqrt{ \left ( -1+x \right ) ^{3}}}{ \left ( -1+x \right ) ^{3/2}}}}\sqrt{-1+x} \right ){\frac{1}{\sqrt{{\frac{\sqrt{ \left ( -1+x \right ) ^{3}}}{ \left ( -1+x \right ) ^{3/2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.41734, size = 11, normalized size = 0.16 \[ 2 \, \arctan \left (\sqrt{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x - 1))

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Fricas [A]  time = 0.268465, size = 11, normalized size = 0.16 \[ 2 \, \arctan \left (\sqrt{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263084, size = 11, normalized size = 0.16 \[ 2 \, \arctan \left (\sqrt{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x - 1))