Optimal. Leaf size=43 \[ \frac{3 \sqrt{x^2-1} x}{8 \left (4-3 x^2\right )}+\frac{5}{16} \tanh ^{-1}\left (\frac{x}{2 \sqrt{x^2-1}}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0379042, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{3 \sqrt{x^2-1} x}{8 \left (4-3 x^2\right )}+\frac{5}{16} \tanh ^{-1}\left (\frac{x}{2 \sqrt{x^2-1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 + x^2]*(-4 + 3*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.02935, size = 34, normalized size = 0.79 \[ \frac{3 x \sqrt{x^{2} - 1}}{8 \left (- 3 x^{2} + 4\right )} + \frac{5 \operatorname{atanh}{\left (\frac{x}{2 \sqrt{x^{2} - 1}} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(3*x**2-4)**2/(x**2-1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0422125, size = 43, normalized size = 1. \[ \frac{5}{16} \tanh ^{-1}\left (\frac{x}{2 \sqrt{x^2-1}}\right )-\frac{3 x \sqrt{x^2-1}}{8 \left (3 x^2-4\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 + x^2]*(-4 + 3*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.069, size = 172, normalized size = 4. \[ -{\frac{1}{16}\sqrt{ \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) ^{2}+{\frac{4\,\sqrt{3}}{3} \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) }+{\frac{1}{3}}} \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) ^{-1}}+{\frac{5}{32}{\it Artanh} \left ({\frac{3\,\sqrt{3}}{2} \left ({\frac{2}{3}}+{\frac{4\,\sqrt{3}}{3} \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) } \right ){\frac{1}{\sqrt{9\, \left ( x-2/3\,\sqrt{3} \right ) ^{2}+12\,\sqrt{3} \left ( x-2/3\,\sqrt{3} \right ) +3}}}} \right ) }-{\frac{1}{16}\sqrt{ \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) ^{2}-{\frac{4\,\sqrt{3}}{3} \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) }+{\frac{1}{3}}} \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) ^{-1}}-{\frac{5}{32}{\it Artanh} \left ({\frac{3\,\sqrt{3}}{2} \left ({\frac{2}{3}}-{\frac{4\,\sqrt{3}}{3} \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) } \right ){\frac{1}{\sqrt{9\, \left ( x+2/3\,\sqrt{3} \right ) ^{2}-12\,\sqrt{3} \left ( x+2/3\,\sqrt{3} \right ) +3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(3*x^2-4)^2/(x^2-1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 4\right )}^{2} \sqrt{x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 4)^2*sqrt(x^2 - 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.21575, size = 201, normalized size = 4.67 \[ \frac{20 \, x^{2} - 5 \,{\left (6 \, x^{4} - 11 \, x^{2} - 2 \,{\left (3 \, x^{3} - 4 \, x\right )} \sqrt{x^{2} - 1} + 4\right )} \log \left (3 \, x^{2} - 3 \, \sqrt{x^{2} - 1} x - 2\right ) + 5 \,{\left (6 \, x^{4} - 11 \, x^{2} - 2 \,{\left (3 \, x^{3} - 4 \, x\right )} \sqrt{x^{2} - 1} + 4\right )} \log \left (x^{2} - \sqrt{x^{2} - 1} x - 2\right ) - 20 \, \sqrt{x^{2} - 1} x - 16}{32 \,{\left (6 \, x^{4} - 11 \, x^{2} - 2 \,{\left (3 \, x^{3} - 4 \, x\right )} \sqrt{x^{2} - 1} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 4)^2*sqrt(x^2 - 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \left (3 x^{2} - 4\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(3*x**2-4)**2/(x**2-1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.23251, size = 127, normalized size = 2.95 \[ \frac{5 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 3}{4 \,{\left (3 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{4} - 10 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 3\right )}} - \frac{5}{32} \,{\rm ln}\left ({\left | 3 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 1 \right |}\right ) + \frac{5}{32} \,{\rm ln}\left ({\left |{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 4)^2*sqrt(x^2 - 1)),x, algorithm="giac")
[Out]