\(\int \frac {1-x^2}{a-b (1-x^2)^4} \, dx\) [11]

Optimal result

Integrand size = 24, antiderivative size = 490 \[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}\right )}{4 \sqrt {a} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}} b^{3/8}}+\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}}-\sqrt {2} \sqrt [8]{b} x}{\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} b^{3/8}}-\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}}+\sqrt {2} \sqrt [8]{b} x}{\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} b^{3/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b}}}\right )}{4 \sqrt {a} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}} b^{3/8}}+\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}} \sqrt [8]{b} x}{\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b} x^2}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} b^{3/8}} \] Output:

1/4*arctan(b^(1/8)*x/(a^(1/4)-b^(1/4))^(1/2))/a^(1/2)/(a^(1/4)-b^(1/4))^(1 
/2)/b^(3/8)+1/8*((a^(1/2)+b^(1/2))^(1/2)-b^(1/4))^(1/2)*arctan((((a^(1/2)+ 
b^(1/2))^(1/2)+b^(1/4))^(1/2)-2^(1/2)*b^(1/8)*x)/((a^(1/2)+b^(1/2))^(1/2)- 
b^(1/4))^(1/2))*2^(1/2)/a^(1/2)/(a^(1/2)+b^(1/2))^(1/2)/b^(3/8)-1/8*((a^(1 
/2)+b^(1/2))^(1/2)-b^(1/4))^(1/2)*arctan((((a^(1/2)+b^(1/2))^(1/2)+b^(1/4) 
)^(1/2)+2^(1/2)*b^(1/8)*x)/((a^(1/2)+b^(1/2))^(1/2)-b^(1/4))^(1/2))*2^(1/2 
)/a^(1/2)/(a^(1/2)+b^(1/2))^(1/2)/b^(3/8)-1/4*arctanh(b^(1/8)*x/(a^(1/4)+b 
^(1/4))^(1/2))/a^(1/2)/(a^(1/4)+b^(1/4))^(1/2)/b^(3/8)+1/8*((a^(1/2)+b^(1/ 
2))^(1/2)+b^(1/4))^(1/2)*arctanh(2^(1/2)*((a^(1/2)+b^(1/2))^(1/2)+b^(1/4)) 
^(1/2)*b^(1/8)*x/((a^(1/2)+b^(1/2))^(1/2)+b^(1/4)*x^2))*2^(1/2)/a^(1/2)/(a 
^(1/2)+b^(1/2))^(1/2)/b^(3/8)
 

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.13 \[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\frac {\text {RootSum}\left [a-b+4 b \text {$\#$1}^2-6 b \text {$\#$1}^4+4 b \text {$\#$1}^6-b \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{8 b} \] Input:

Integrate[(1 - x^2)/(a - b*(1 - x^2)^4),x]
 

Output:

RootSum[a - b + 4*b*#1^2 - 6*b*#1^4 + 4*b*#1^6 - b*#1^8 & , Log[x - #1]/(# 
1 - 2*#1^3 + #1^5) & ]/(8*b)
 

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7291, 2009}

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.

\(\displaystyle \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx\)

\(\Big \downarrow \) 7291

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}\right )}{4 \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b}}}+\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}}-\sqrt {2} \sqrt [8]{b} x}{\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}}}\right )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}}+\sqrt {2} \sqrt [8]{b} x}{\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}-\sqrt [4]{b}}}\right )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b}}}\right )}{4 \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{a}+\sqrt [4]{b}}}-\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}} \log \left (-\sqrt {2} \sqrt [8]{b} x \sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}}+\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}} \log \left (\sqrt {2} \sqrt [8]{b} x \sqrt {\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b}}+\sqrt {\sqrt {a}+\sqrt {b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {a}+\sqrt {b}}}\)

Input:

Int[(1 - x^2)/(a - b*(1 - x^2)^4),x]
 

Output:

ArcTan[(b^(1/8)*x)/Sqrt[a^(1/4) - b^(1/4)]]/(4*Sqrt[a]*Sqrt[a^(1/4) - b^(1 
/4)]*b^(3/8)) + (Sqrt[Sqrt[Sqrt[a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt 
[Sqrt[a] + Sqrt[b]] + b^(1/4)] - Sqrt[2]*b^(1/8)*x)/Sqrt[Sqrt[Sqrt[a] + Sq 
rt[b]] - b^(1/4)]])/(4*Sqrt[2]*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/8)) - 
(Sqrt[Sqrt[Sqrt[a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt[Sqrt[a] + Sqrt[ 
b]] + b^(1/4)] + Sqrt[2]*b^(1/8)*x)/Sqrt[Sqrt[Sqrt[a] + Sqrt[b]] - b^(1/4) 
]])/(4*Sqrt[2]*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/8)) - ArcTanh[(b^(1/8) 
*x)/Sqrt[a^(1/4) + b^(1/4)]]/(4*Sqrt[a]*Sqrt[a^(1/4) + b^(1/4)]*b^(3/8)) - 
 (Sqrt[Sqrt[Sqrt[a] + Sqrt[b]] + b^(1/4)]*Log[Sqrt[Sqrt[a] + Sqrt[b]] - Sq 
rt[2]*Sqrt[Sqrt[Sqrt[a] + Sqrt[b]] + b^(1/4)]*b^(1/8)*x + b^(1/4)*x^2])/(8 
*Sqrt[2]*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/8)) + (Sqrt[Sqrt[Sqrt[a] + S 
qrt[b]] + b^(1/4)]*Log[Sqrt[Sqrt[a] + Sqrt[b]] + Sqrt[2]*Sqrt[Sqrt[Sqrt[a] 
 + Sqrt[b]] + b^(1/4)]*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*Sqrt[a]*Sqrt[S 
qrt[a] + Sqrt[b]]*b^(3/8))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7291

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[Polyno 
mialInSubst[v, u, x]/(a + b*x^n), x] /. x -> u, x] /; FreeQ[{a, b}, x] && I 
GtQ[n, 0] && PolynomialInQ[v, u, x]
 

Maple [C] (verified)

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

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.14

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}-4 \textit {\_Z}^{6} b +6 b \,\textit {\_Z}^{4}-4 b \,\textit {\_Z}^{2}-a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5}-3 \textit {\_R}^{3}+\textit {\_R}}}{8 b}\)	\(71\)
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}-4 \textit {\_Z}^{6} b +6 b \,\textit {\_Z}^{4}-4 b \,\textit {\_Z}^{2}-a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5}-3 \textit {\_R}^{3}+\textit {\_R}}}{8 b}\)	\(71\)

Input:

int((-x^2+1)/(a-b*(-x^2+1)^4),x,method=_RETURNVERBOSE)
 

Output:

1/8/b*sum((-_R^2+1)/(-_R^7+3*_R^5-3*_R^3+_R)*ln(x-_R),_R=RootOf(_Z^8*b-4*_ 
Z^6*b+6*_Z^4*b-4*_Z^2*b-a+b))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\text {Timed out} \] Input:

integrate((-x^2+1)/(a-b*(-x^2+1)^4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [A] (verification not implemented)

Time = 2.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.27 \[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\operatorname {RootSum} {\left (t^{8} \cdot \left (16777216 a^{5} b^{3} - 16777216 a^{4} b^{4}\right ) + 1048576 t^{6} a^{3} b^{3} - 24576 t^{4} a^{2} b^{2} + 256 t^{2} a b - 1, \left ( t \mapsto t \log {\left (- 6291456 t^{7} a^{4} b^{3} + 6291456 t^{7} a^{3} b^{4} - 65536 t^{5} a^{3} b^{2} - 327680 t^{5} a^{2} b^{3} - 512 t^{3} a^{2} b + 5632 t^{3} a b^{2} - 32 t b + x \right )} \right )\right )} \] Input:

integrate((-x**2+1)/(a-b*(-x**2+1)**4),x)
 

Output:

RootSum(_t**8*(16777216*a**5*b**3 - 16777216*a**4*b**4) + 1048576*_t**6*a* 
*3*b**3 - 24576*_t**4*a**2*b**2 + 256*_t**2*a*b - 1, Lambda(_t, _t*log(-62 
91456*_t**7*a**4*b**3 + 6291456*_t**7*a**3*b**4 - 65536*_t**5*a**3*b**2 - 
327680*_t**5*a**2*b**3 - 512*_t**3*a**2*b + 5632*_t**3*a*b**2 - 32*_t*b + 
x)))
 

Maxima [F]

\[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{2} - 1\right )}^{4} b - a} \,d x } \] Input:

integrate((-x^2+1)/(a-b*(-x^2+1)^4),x, algorithm="maxima")
 

Output:

integrate((x^2 - 1)/((x^2 - 1)^4*b - a), x)
 

Giac [F]

\[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{2} - 1\right )}^{4} b - a} \,d x } \] Input:

integrate((-x^2+1)/(a-b*(-x^2+1)^4),x, algorithm="giac")
 

Output:

integrate((x^2 - 1)/((x^2 - 1)^4*b - a), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 10.24 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.67 \[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\sum _{k=1}^8\ln \left (-a\,b^5\,\left ({\mathrm {root}\left (16777216\,a^5\,b^3\,z^8-16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6-24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2-1,z,k\right )}^2\,a\,b\,64-1\right )\,\left ({\mathrm {root}\left (16777216\,a^5\,b^3\,z^8-16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6-24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2-1,z,k\right )}^4\,a^2\,b^2\,4096-{\mathrm {root}\left (16777216\,a^5\,b^3\,z^8-16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6-24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2-1,z,k\right )}^2\,a\,b\,128-{\mathrm {root}\left (16777216\,a^5\,b^3\,z^8-16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6-24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2-1,z,k\right )}^5\,a^3\,b^2\,x\,32768+1\right )\right )\,\mathrm {root}\left (16777216\,a^5\,b^3\,z^8-16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6-24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2-1,z,k\right ) \] Input:

int(-(x^2 - 1)/(a - b*(x^2 - 1)^4),x)
 

Output:

symsum(log(-a*b^5*(64*root(16777216*a^5*b^3*z^8 - 16777216*a^4*b^4*z^8 + 1 
048576*a^3*b^3*z^6 - 24576*a^2*b^2*z^4 + 256*a*b*z^2 - 1, z, k)^2*a*b - 1) 
*(4096*root(16777216*a^5*b^3*z^8 - 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3* 
z^6 - 24576*a^2*b^2*z^4 + 256*a*b*z^2 - 1, z, k)^4*a^2*b^2 - 128*root(1677 
7216*a^5*b^3*z^8 - 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3*z^6 - 24576*a^2* 
b^2*z^4 + 256*a*b*z^2 - 1, z, k)^2*a*b - 32768*root(16777216*a^5*b^3*z^8 - 
 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3*z^6 - 24576*a^2*b^2*z^4 + 256*a*b* 
z^2 - 1, z, k)^5*a^3*b^2*x + 1))*root(16777216*a^5*b^3*z^8 - 16777216*a^4* 
b^4*z^8 + 1048576*a^3*b^3*z^6 - 24576*a^2*b^2*z^4 + 256*a*b*z^2 - 1, z, k) 
, k, 1, 8)
 

Reduce [F]

\[ \int \frac {1-x^2}{a-b \left (1-x^2\right )^4} \, dx=\int \frac {-x^{2}+1}{a -b \left (-x^{2}+1\right )^{4}}d x \] Input:

int((-x^2+1)/(a-b*(-x^2+1)^4),x)
 

Output:

int((-x^2+1)/(a-b*(-x^2+1)^4),x)