Optimal. Leaf size=45 \[ \frac{1}{8} \sqrt{4 x^4-3 x^2}+\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4-3 x^2}}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.101116, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{1}{8} \sqrt{4 x^4-3 x^2}+\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4-3 x^2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^3/Sqrt[-3*x^2 + 4*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.7373, size = 37, normalized size = 0.82 \[ \frac{\sqrt{4 x^{4} - 3 x^{2}}}{8} + \frac{3 \operatorname{atanh}{\left (\frac{2 x^{2}}{\sqrt{4 x^{4} - 3 x^{2}}} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(4*x**4-3*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0214401, size = 58, normalized size = 1.29 \[ \frac{x \left (8 x^3+3 \sqrt{4 x^2-3} \log \left (\sqrt{4 x^2-3}+2 x\right )-6 x\right )}{16 \sqrt{x^2 \left (4 x^2-3\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/Sqrt[-3*x^2 + 4*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 60, normalized size = 1.3 \[{\frac{x}{32}\sqrt{4\,{x}^{2}-3} \left ( 3\,\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-3} \right ) \sqrt{4}+4\,x\sqrt{4\,{x}^{2}-3} \right ){\frac{1}{\sqrt{4\,{x}^{4}-3\,{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(4*x^4-3*x^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.764865, size = 55, normalized size = 1.22 \[ \frac{1}{8} \, \sqrt{4 \, x^{4} - 3 \, x^{2}} + \frac{3}{32} \, \log \left (8 \, x^{2} + 4 \, \sqrt{4 \, x^{4} - 3 \, x^{2}} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(4*x^4 - 3*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.261799, size = 61, normalized size = 1.36 \[ \frac{1}{8} \, \sqrt{4 \, x^{4} - 3 \, x^{2}} - \frac{3}{16} \, \log \left (-\frac{2 \, x^{2} - \sqrt{4 \, x^{4} - 3 \, x^{2}}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(4*x^4 - 3*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{2} \left (4 x^{2} - 3\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(4*x**4-3*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.282338, size = 57, normalized size = 1.27 \[ \frac{1}{8} \, \sqrt{4 \, x^{4} - 3 \, x^{2}} - \frac{3}{32} \,{\rm ln}\left ({\left | -8 \, x^{2} + 4 \, \sqrt{4 \, x^{4} - 3 \, x^{2}} + 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(4*x^4 - 3*x^2),x, algorithm="giac")
[Out]