3.21 \(\int \frac{5+2 x^2}{-1+x^4} \, dx\)

Optimal. Leaf size=13 \[ -\frac{3}{2} \tan ^{-1}(x)-\frac{7}{2} \tanh ^{-1}(x) \]

[Out]

(-3*ArcTan[x])/2 - (7*ArcTanh[x])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0178989, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3}{2} \tan ^{-1}(x)-\frac{7}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(5 + 2*x^2)/(-1 + x^4),x]

[Out]

(-3*ArcTan[x])/2 - (7*ArcTanh[x])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.45043, size = 14, normalized size = 1.08 \[ - \frac{3 \operatorname{atan}{\left (x \right )}}{2} - \frac{7 \operatorname{atanh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2+5)/(x**4-1),x)

[Out]

-3*atan(x)/2 - 7*atanh(x)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00909264, size = 25, normalized size = 1.92 \[ \frac{7}{4} \log (1-x)-\frac{7}{4} \log (x+1)-\frac{3}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + 2*x^2)/(-1 + x^4),x]

[Out]

(-3*ArcTan[x])/2 + (7*Log[1 - x])/4 - (7*Log[1 + x])/4

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 18, normalized size = 1.4 \[{\frac{7\,\ln \left ( -1+x \right ) }{4}}-{\frac{7\,\ln \left ( 1+x \right ) }{4}}-{\frac{3\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+5)/(x^4-1),x)

[Out]

7/4*ln(-1+x)-7/4*ln(1+x)-3/2*arctan(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.860507, size = 23, normalized size = 1.77 \[ -\frac{3}{2} \, \arctan \left (x\right ) - \frac{7}{4} \, \log \left (x + 1\right ) + \frac{7}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 5)/(x^4 - 1),x, algorithm="maxima")

[Out]

-3/2*arctan(x) - 7/4*log(x + 1) + 7/4*log(x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.277699, size = 23, normalized size = 1.77 \[ -\frac{3}{2} \, \arctan \left (x\right ) - \frac{7}{4} \, \log \left (x + 1\right ) + \frac{7}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 5)/(x^4 - 1),x, algorithm="fricas")

[Out]

-3/2*arctan(x) - 7/4*log(x + 1) + 7/4*log(x - 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.418141, size = 22, normalized size = 1.69 \[ \frac{7 \log{\left (x - 1 \right )}}{4} - \frac{7 \log{\left (x + 1 \right )}}{4} - \frac{3 \operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2+5)/(x**4-1),x)

[Out]

7*log(x - 1)/4 - 7*log(x + 1)/4 - 3*atan(x)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269146, size = 26, normalized size = 2. \[ -\frac{3}{2} \, \arctan \left (x\right ) - \frac{7}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{7}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 5)/(x^4 - 1),x, algorithm="giac")

[Out]

-3/2*arctan(x) - 7/4*ln(abs(x + 1)) + 7/4*ln(abs(x - 1))