\(\int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx\) [831]

Optimal result

Integrand size = 22, antiderivative size = 114 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=\frac {x}{a}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^{3/2} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{5/4} \sqrt {b}}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^{3/2} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{5/4} \sqrt {b}} \] Output:

x/a+1/2*(a^(1/2)-b^(1/2))^(3/2)*arctan(a^(1/4)*x/(a^(1/2)-b^(1/2))^(1/2))/ 
a^(5/4)/b^(1/2)-1/2*(a^(1/2)+b^(1/2))^(3/2)*arctan(a^(1/4)*x/(a^(1/2)+b^(1 
/2))^(1/2))/a^(5/4)/b^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.26 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=\frac {x}{a}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {a-\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:

Integrate[x^4/(a - b + 2*a*x^2 + a*x^4),x]
 

Output:

x/a + ((Sqrt[a] - Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]] 
)/(2*a*Sqrt[a - Sqrt[a]*Sqrt[b]]*Sqrt[b]) - ((Sqrt[a] + Sqrt[b])^2*ArcTan[ 
(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(2*a*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqr 
t[b])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1442, 1480, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^4}{a x^4+2 a x^2+a-b} \, dx\)

\(\Big \downarrow \) 1442

\(\displaystyle \frac {x}{a}-\frac {\int \frac {2 a x^2+a-b}{a x^4+2 a x^2+a-b}dx}{a}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {x}{a}-\frac {\frac {1}{2} \sqrt {a} \left (2 \sqrt {a}-\frac {a+b}{\sqrt {b}}\right ) \int \frac {1}{a x^2+\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}dx+\frac {1}{2} \sqrt {a} \left (\frac {a+b}{\sqrt {b}}+2 \sqrt {a}\right ) \int \frac {1}{a x^2+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}dx}{a}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {x}{a}-\frac {\frac {\left (2 \sqrt {a}-\frac {a+b}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (\frac {a+b}{\sqrt {b}}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}}}}{a}\)

Input:

Int[x^4/(a - b + 2*a*x^2 + a*x^4),x]
 

Output:

x/a - (((2*Sqrt[a] - (a + b)/Sqrt[b])*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sq 
rt[b]]])/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + ((2*Sqrt[a] + (a + b)/Sqrt[ 
b])*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(1/4)*Sqrt[Sqrt[a] + 
 Sqrt[b]]))/a
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50

method	result	size
risch	\(\frac {x}{a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (-2 \textit {\_R}^{2} a -a +b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a^{2}}\)	\(57\)
default	\(\frac {x}{a}-\frac {\left (-2 \sqrt {a b}+a +b \right ) \operatorname {arctanh}\left (\frac {a x}{\sqrt {\left (\sqrt {a b}-a \right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) a}}+\frac {\left (-2 \sqrt {a b}-a -b \right ) \arctan \left (\frac {a x}{\sqrt {\left (\sqrt {a b}+a \right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) a}}\)	\(101\)

Input:

int(x^4/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)
 

Output:

x/a+1/4/a^2*sum((-2*_R^2*a-a+b)/(_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4*a+2*_Z^2 
*a+a-b))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (74) = 148\).

Time = 0.08 (sec) , antiderivative size = 603, normalized size of antiderivative = 5.29 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=\frac {a \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b - a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}}\right ) - a \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b - a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}}\right ) - a \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b + a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}}\right ) + a \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b + a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}}\right ) + 4 \, x}{4 \, a} \] Input:

integrate(x^4/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
 

Output:

1/4*(a*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + a + 3*b)/(a^2*b) 
)*log(-(3*a^2 - 2*a*b - b^2)*x + (a^4*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b) 
) - 3*a^2*b - a*b^2)*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + a 
+ 3*b)/(a^2*b))) - a*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + a 
+ 3*b)/(a^2*b))*log(-(3*a^2 - 2*a*b - b^2)*x - (a^4*b*sqrt((9*a^2 + 6*a*b 
+ b^2)/(a^5*b)) - 3*a^2*b - a*b^2)*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2) 
/(a^5*b)) + a + 3*b)/(a^2*b))) - a*sqrt((a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/ 
(a^5*b)) - a - 3*b)/(a^2*b))*log(-(3*a^2 - 2*a*b - b^2)*x + (a^4*b*sqrt((9 
*a^2 + 6*a*b + b^2)/(a^5*b)) + 3*a^2*b + a*b^2)*sqrt((a^2*b*sqrt((9*a^2 + 
6*a*b + b^2)/(a^5*b)) - a - 3*b)/(a^2*b))) + a*sqrt((a^2*b*sqrt((9*a^2 + 6 
*a*b + b^2)/(a^5*b)) - a - 3*b)/(a^2*b))*log(-(3*a^2 - 2*a*b - b^2)*x - (a 
^4*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + 3*a^2*b + a*b^2)*sqrt((a^2*b*sq 
rt((9*a^2 + 6*a*b + b^2)/(a^5*b)) - a - 3*b)/(a^2*b))) + 4*x)/a
 

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{5} b^{2} + t^{2} \cdot \left (32 a^{4} b + 96 a^{3} b^{2}\right ) + a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} b + 4 t a^{3} + 24 t a^{2} b + 4 t a b^{2}}{3 a^{2} - 2 a b - b^{2}} \right )} \right )\right )} + \frac {x}{a} \] Input:

integrate(x**4/(a*x**4+2*a*x**2+a-b),x)
 

Output:

RootSum(256*_t**4*a**5*b**2 + _t**2*(32*a**4*b + 96*a**3*b**2) + a**3 - 3* 
a**2*b + 3*a*b**2 - b**3, Lambda(_t, _t*log(x + (64*_t**3*a**4*b + 4*_t*a* 
*3 + 24*_t*a**2*b + 4*_t*a*b**2)/(3*a**2 - 2*a*b - b**2)))) + x/a
 

Maxima [F]

\[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=\int { \frac {x^{4}}{a x^{4} + 2 \, a x^{2} + a - b} \,d x } \] Input:

integrate(x^4/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")
 

Output:

x/a - integrate((2*a*x^2 + a - b)/(a*x^4 + 2*a*x^2 + a - b), x)/a
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (74) = 148\).

Time = 0.17 (sec) , antiderivative size = 511, normalized size of antiderivative = 4.48 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=-\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{4} - \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{3} b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} b^{2} - 2 \, {\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} b^{2}\right )} a^{2} + {\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} a^{3} b - 7 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} + \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b - 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{4} - \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{3} b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} b^{2} - 2 \, {\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} b^{2}\right )} a^{2} - {\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} a^{3} b - 7 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} - \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b - 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} + \frac {x}{a} \] Input:

integrate(x^4/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")
 

Output:

-1/2*(3*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^4 - sqrt(a^2 + sqrt(a*b)*a)*sq 
rt(a*b)*a^3*b - 4*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^2*b^2 - 2*(3*sqrt(a^ 
2 + sqrt(a*b)*a)*sqrt(a*b)*a*b - 4*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*b^2)* 
a^2 + (3*sqrt(a^2 + sqrt(a*b)*a)*a^3*b - 7*sqrt(a^2 + sqrt(a*b)*a)*a^2*b^2 
 + 4*sqrt(a^2 + sqrt(a*b)*a)*a*b^3)*abs(a))*arctan(x/sqrt((a^2 + sqrt(a^4 
- (a^2 - a*b)*a^2))/a^2))/(3*a^6*b - 7*a^5*b^2 + 4*a^4*b^3) + 1/2*(3*sqrt( 
a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^4 - sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^3*b 
 - 4*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^2*b^2 - 2*(3*sqrt(a^2 - sqrt(a*b) 
*a)*sqrt(a*b)*a*b - 4*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*b^2)*a^2 - (3*sqrt 
(a^2 - sqrt(a*b)*a)*a^3*b - 7*sqrt(a^2 - sqrt(a*b)*a)*a^2*b^2 + 4*sqrt(a^2 
 - sqrt(a*b)*a)*a*b^3)*abs(a))*arctan(x/sqrt((a^2 - sqrt(a^4 - (a^2 - a*b) 
*a^2))/a^2))/(3*a^6*b - 7*a^5*b^2 + 4*a^4*b^3) + x/a
 

Mupad [B] (verification not implemented)

Time = 19.01 (sec) , antiderivative size = 1097, normalized size of antiderivative = 9.62 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx =\text {Too large to display} \] Input:

int(x^4/(a - b + 2*a*x^2 + a*x^4),x)
 

Output:

x/a - 2*atanh((24*x*(a^5*b^3)^(1/2)*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a^5*b 
^3)^(1/2))/(16*a^4*b^2) - (a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b^2 - (6 
*(a^5*b^3)^(1/2))/a - 6*a^2*b + 2*b^3 + (2*b^2*(a^5*b^3)^(1/2))/a^3 + (4*b 
*(a^5*b^3)^(1/2))/a^2) + (8*x*(a^5*b^3)^(1/2)*(- 3/(16*a^2) - 1/(16*a*b) - 
 (3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - (a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((4 
*(a^5*b^3)^(1/2))/a - (6*(a^5*b^3)^(1/2))/b + 2*a*b^2 + 4*a^2*b - 6*a^3 + 
(2*b*(a^5*b^3)^(1/2))/a^2) - (8*a*b^2*x*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a 
^5*b^3)^(1/2))/(16*a^4*b^2) - (a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b + 
(4*(a^5*b^3)^(1/2))/a^2 - 6*a^2 + 2*b^2 - (6*(a^5*b^3)^(1/2))/(a*b) + (2*b 
*(a^5*b^3)^(1/2))/a^3) - (24*a^2*b*x*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a^5* 
b^3)^(1/2))/(16*a^4*b^2) - (a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b + (4* 
(a^5*b^3)^(1/2))/a^2 - 6*a^2 + 2*b^2 - (6*(a^5*b^3)^(1/2))/(a*b) + (2*b*(a 
^5*b^3)^(1/2))/a^3))*(-(3*a*(a^5*b^3)^(1/2) + b*(a^5*b^3)^(1/2) + a^4*b + 
3*a^3*b^2)/(16*a^5*b^2))^(1/2) + 2*atanh((24*x*(a^5*b^3)^(1/2)*((3*(a^5*b^ 
3)^(1/2))/(16*a^4*b^2) - 1/(16*a*b) - 3/(16*a^2) + (a^5*b^3)^(1/2)/(16*a^5 
*b))^(1/2))/((6*(a^5*b^3)^(1/2))/a + 4*a*b^2 - 6*a^2*b + 2*b^3 - (2*b^2*(a 
^5*b^3)^(1/2))/a^3 - (4*b*(a^5*b^3)^(1/2))/a^2) - (8*x*(a^5*b^3)^(1/2)*((3 
*(a^5*b^3)^(1/2))/(16*a^4*b^2) - 1/(16*a*b) - 3/(16*a^2) + (a^5*b^3)^(1/2) 
/(16*a^5*b))^(1/2))/((4*(a^5*b^3)^(1/2))/a - (6*(a^5*b^3)^(1/2))/b - 2*a*b 
^2 - 4*a^2*b + 6*a^3 + (2*b*(a^5*b^3)^(1/2))/a^2) + (8*a*b^2*x*((3*(a^5...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.77 \[ \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}\, \mathit {atan} \left (\frac {a x}{\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}}\right ) b -2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}\, \mathit {atan} \left (\frac {a x}{\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}}\right ) a +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a +4 a b x}{4 a^{2} b} \] Input:

int(x^4/(a*x^4+2*a*x^2+a-b),x)
 

Output:

( - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a) + a)*atan((a*x)/(sqrt(a)*sqrt(sqrt(b)*s 
qrt(a) + a)))*b - 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a) + a)*atan((a*x)/(sqrt(a)* 
sqrt(sqrt(b)*sqrt(a) + a)))*a + sqrt(a)*sqrt(sqrt(b)*sqrt(a) - a)*log( - s 
qrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b - sqrt(a)*sqrt(sqrt(b)*sqrt(a) - a 
)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b - sqrt(b)*sqrt(sqrt(b)*sqrt 
(a) - a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a + sqrt(b)*sqrt(sq 
rt(b)*sqrt(a) - a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a + 4*a*b*x) 
/(4*a**2*b)