Optimal. Leaf size=56 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}}+\frac{\log \left (a x^4+2 a x^2+a-b\right )}{4 a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.107204, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}}+\frac{\log \left (a x^4+2 a x^2+a-b\right )}{4 a} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a - b + 2*a*x^2 + a*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.7315, size = 48, normalized size = 0.86 \[ \frac{\log{\left (a x^{4} + 2 a x^{2} + a - b \right )}}{4 a} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{a} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a*x**4+2*a*x**2+a-b),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0300339, size = 51, normalized size = 0.91 \[ \frac{\log \left (a \left (x^2+1\right )^2-b\right )+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{\sqrt{b}}}{4 a} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a - b + 2*a*x^2 + a*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.004, size = 49, normalized size = 0.9 \[{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a-b \right ) }{4\,a}}+{\frac{1}{2}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a*x^4+2*a*x^2+a-b),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.281923, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (\frac{2 \, a b x^{2} + 2 \, a b +{\left (a x^{4} + 2 \, a x^{2} + a + b\right )} \sqrt{a b}}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + \sqrt{a b} \log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, \sqrt{a b} a}, -\frac{2 \, a \arctan \left (\frac{b}{\sqrt{-a b}{\left (x^{2} + 1\right )}}\right ) - \sqrt{-a b} \log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, \sqrt{-a b} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.47693, size = 110, normalized size = 1.96 \[ \left (\frac{1}{4 a} - \frac{\sqrt{a^{3} b}}{4 a^{2} b}\right ) \log{\left (x^{2} + \frac{4 a b \left (\frac{1}{4 a} - \frac{\sqrt{a^{3} b}}{4 a^{2} b}\right ) + a - b}{a} \right )} + \left (\frac{1}{4 a} + \frac{\sqrt{a^{3} b}}{4 a^{2} b}\right ) \log{\left (x^{2} + \frac{4 a b \left (\frac{1}{4 a} + \frac{\sqrt{a^{3} b}}{4 a^{2} b}\right ) + a - b}{a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a*x**4+2*a*x**2+a-b),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.548466, size = 62, normalized size = 1.11 \[ -\frac{\arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b}} + \frac{{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="giac")
[Out]