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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]