3.4 \(\int \frac{1}{\sqrt{-1+x^2} \left (-4+3 x^2\right )^2} \, dx\)

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]

(3*x*Sqrt[-1 + x^2])/(8*(4 - 3*x^2)) + (5*ArcTanh[x/(2*Sqrt[-1 + x^2])])/16

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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]

(3*x*Sqrt[-1 + x^2])/(8*(4 - 3*x^2)) + (5*ArcTanh[x/(2*Sqrt[-1 + x^2])])/16

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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]

3*x*sqrt(x**2 - 1)/(8*(-3*x**2 + 4)) + 5*atanh(x/(2*sqrt(x**2 - 1)))/16

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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]

(-3*x*Sqrt[-1 + x^2])/(8*(-4 + 3*x^2)) + (5*ArcTanh[x/(2*Sqrt[-1 + x^2])])/16

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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]

-1/16/(x-2/3*3^(1/2))*((x-2/3*3^(1/2))^2+4/3*3^(1/2)*(x-2/3*3^(1/2))+1/3)^(1/2)+
5/32*arctanh(3/2*(2/3+4/3*3^(1/2)*(x-2/3*3^(1/2)))*3^(1/2)/(9*(x-2/3*3^(1/2))^2+
12*3^(1/2)*(x-2/3*3^(1/2))+3)^(1/2))-1/16/(x+2/3*3^(1/2))*((x+2/3*3^(1/2))^2-4/3
*3^(1/2)*(x+2/3*3^(1/2))+1/3)^(1/2)-5/32*arctanh(3/2*(2/3-4/3*3^(1/2)*(x+2/3*3^(
1/2)))*3^(1/2)/(9*(x+2/3*3^(1/2))^2-12*3^(1/2)*(x+2/3*3^(1/2))+3)^(1/2))

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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]

integrate(1/((3*x^2 - 4)^2*sqrt(x^2 - 1)), x)

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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]

1/32*(20*x^2 - 5*(6*x^4 - 11*x^2 - 2*(3*x^3 - 4*x)*sqrt(x^2 - 1) + 4)*log(3*x^2
- 3*sqrt(x^2 - 1)*x - 2) + 5*(6*x^4 - 11*x^2 - 2*(3*x^3 - 4*x)*sqrt(x^2 - 1) + 4
)*log(x^2 - sqrt(x^2 - 1)*x - 2) - 20*sqrt(x^2 - 1)*x - 16)/(6*x^4 - 11*x^2 - 2*
(3*x^3 - 4*x)*sqrt(x^2 - 1) + 4)

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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]

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

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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]

1/4*(5*(x - sqrt(x^2 - 1))^2 - 3)/(3*(x - sqrt(x^2 - 1))^4 - 10*(x - sqrt(x^2 -
1))^2 + 3) - 5/32*ln(abs(3*(x - sqrt(x^2 - 1))^2 - 1)) + 5/32*ln(abs((x - sqrt(x
^2 - 1))^2 - 3))