3.665 \(\int \frac{\sqrt{x}-\sqrt{x^3}}{x-x^3} \, dx\)

Optimal. Leaf size=52 \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.287317, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.166167, size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}-\sqrt{x^3}}{x-x^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(Sqrt[x] - Sqrt[x^3])/(x - x^3), x]

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 41, normalized size = 0.8 \[{\it Artanh} \left ( \sqrt{x} \right ) +\arctan \left ( \sqrt{x} \right ) +{\frac{1}{2}\sqrt{{x}^{3}} \left ( \ln \left ( -1+\sqrt{x} \right ) -\ln \left ( 1+\sqrt{x} \right ) +2\,\arctan \left ( \sqrt{x} \right ) \right ){x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.852144, size = 58, normalized size = 1.12 \[ 2 \, \arctan \left (\sqrt{x}\right ) - \frac{1}{2} \, \log \left (4 \, \sqrt{x} + 4\right ) + \frac{1}{2} \, \log \left (4 \, \sqrt{x} - 4\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x)) - 1/2*log(4*sqrt(x) + 4) + 1/2*log(4*sqrt(x) - 4) + 1/2*log(sq
rt(x) + 1) - 1/2*log(sqrt(x) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.271974, size = 8, normalized size = 0.15 \[ 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264907, size = 8, normalized size = 0.15 \[ 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x))