Integrand size = 24, antiderivative size = 257 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=-\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a-b) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\text {arctanh}(x)+\frac {\left (\sqrt {a}-\sqrt {b}\right ) (a-b) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) (a-b) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}} \] Output:
1/4*(a^(1/2)+b^(1/2))*(a-b)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a ^(3/4)/b^(3/4)+1/4*(a^(1/2)+b^(1/2))*(a-b)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1 /4))*2^(1/2)/a^(3/4)/b^(3/4)-arctanh(x)+1/8*(a^(1/2)-b^(1/2))*(a-b)*ln(a^( 1/2)-2^(1/2)*a^(1/4)*b^(1/4)*x+b^(1/2)*x^2)*2^(1/2)/a^(3/4)/b^(3/4)-1/8*(a ^(1/2)-b^(1/2))*(a-b)*ln(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*x+b^(1/2)*x^2)*2^ (1/2)/a^(3/4)/b^(3/4)
Time = 0.09 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.02 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=\frac {-2 \sqrt {2} \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right ) (a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right ) (a-b) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+4 a b^{3/4} \log (1-x)-4 a b^{3/4} \log (1+x)+\sqrt {2} \left (a^{3/4}-\sqrt [4]{a} \sqrt {b}\right ) (a-b) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-\sqrt {2} \left (a^{3/4}-\sqrt [4]{a} \sqrt {b}\right ) (a-b) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{8 a b^{3/4}} \] Input:
Integrate[(b + a*x^4)/((-1 + x^2)*(a + b*x^4)),x]
Output:
(-2*Sqrt[2]*a^(1/4)*(Sqrt[a] + Sqrt[b])*(a - b)*ArcTan[1 - (Sqrt[2]*b^(1/4 )*x)/a^(1/4)] + 2*Sqrt[2]*a^(1/4)*(Sqrt[a] + Sqrt[b])*(a - b)*ArcTan[1 + ( Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 4*a*b^(3/4)*Log[1 - x] - 4*a*b^(3/4)*Log[1 + x] + Sqrt[2]*(a^(3/4) - a^(1/4)*Sqrt[b])*(a - b)*Log[Sqrt[a] - Sqrt[2]*a^ (1/4)*b^(1/4)*x + Sqrt[b]*x^2] - Sqrt[2]*(a^(3/4) - a^(1/4)*Sqrt[b])*(a - b)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(8*a*b^(3/4))
Time = 0.46 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2257, 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.
| \(\displaystyle \int \frac {a x^4+b}{\left (x^2-1\right ) \left (a+b x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle \int \left (\frac {\left (x^2+1\right ) (a-b)}{a+b x^4}+\frac {1}{x^2-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a-b) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) (a-b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) (a-b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\text {arctanh}(x)\) |
Input:
Int[(b + a*x^4)/((-1 + x^2)*(a + b*x^4)),x]
Output:
-1/2*((Sqrt[a] + Sqrt[b])*(a - b)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)]) /(Sqrt[2]*a^(3/4)*b^(3/4)) + ((Sqrt[a] + Sqrt[b])*(a - b)*ArcTan[1 + (Sqrt [2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ArcTanh[x] + ((Sqrt [a] - Sqrt[b])*(a - b)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x ^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4)) - ((Sqrt[a] - Sqrt[b])*(a - b)*Log[Sqrt[a ] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4))
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Time = 0.12 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\ln \left (-1+x \right )}{2}+\frac {\left (a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (a -b \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\ln \left (1+x \right )}{2}\) | \(226\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{3} b^{3} \textit {\_Z}^{4}+\left (4 a^{4} b^{2}-8 a^{3} b^{3}+4 a^{2} b^{4}\right ) \textit {\_Z}^{2}+a^{6}-2 a^{5} b -a^{4} b^{2}+4 a^{3} b^{3}-a^{2} b^{4}-2 a \,b^{5}+b^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} b^{3}+2 a^{2} b^{4}\right ) \textit {\_R}^{5}+\left (-7 a^{4} b^{2}+16 a^{3} b^{3}-18 a^{2} b^{4}+b^{6}\right ) \textit {\_R}^{3}+\left (-2 a^{6}+4 a^{5} b -2 a^{4} b^{2}-8 a^{3} b^{3}+10 a^{2} b^{4}+4 a \,b^{5}-6 b^{6}\right ) \textit {\_R} \right ) x +\left (a^{3} b^{3}+4 a^{2} b^{4}-a \,b^{5}\right ) \textit {\_R}^{4}+\left (-a^{5} b +17 a^{4} b^{2}-18 a^{3} b^{3}-2 a^{2} b^{4}+3 a \,b^{5}+b^{6}\right ) \textit {\_R}^{2}+4 a^{6}-4 a^{5} b -8 a^{4} b^{2}+8 a^{3} b^{3}+4 a^{2} b^{4}-4 a \,b^{5}\right )\right )}{4}-\frac {\ln \left (1+x \right )}{2}+\frac {\ln \left (1-x \right )}{2}\) | \(324\) |
Input:
int((a*x^4+b)/(x^2-1)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/2*ln(-1+x)+1/8*(a-b)*(a/b)^(1/4)/a*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2 )+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/( a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1))+1/8*(a-b)/b/(a/b)^(1/4) *2^(1/2)*(ln((x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^ (1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/ b)^(1/4)*x-1))-1/2*ln(1+x)
Leaf count of result is larger than twice the leaf count of optimal. 1179 vs. \(2 (176) = 352\).
Time = 0.12 (sec) , antiderivative size = 1179, normalized size of antiderivative = 4.59 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate((a*x^4+b)/(x^2-1)/(b*x^4+a),x, algorithm="fricas")
Output:
-1/4*sqrt(-(a*b*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^ 4 - 6*a*b^5 + b^6)/(a^3*b^3)) + 2*a^2 - 4*a*b + 2*b^2)/(a*b))*log((a^5 - 3 *a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5)*x + (a^3*b^2*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a^3*b^3)) - a^4*b + 3*a^3*b^2 - 3*a^2*b^3 + a*b^4)*sqrt(-(a*b*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a^3*b^3)) + 2*a^2 - 4*a*b + 2*b^2)/(a*b))) + 1/4*sqrt(-(a*b*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b ^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a^3*b^3)) + 2*a^2 - 4*a*b + 2*b^2)/(a*b))*log((a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5) *x - (a^3*b^2*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a^3*b^3)) - a^4*b + 3*a^3*b^2 - 3*a^2*b^3 + a*b^4)*sqrt( -(a*b*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^ 5 + b^6)/(a^3*b^3)) + 2*a^2 - 4*a*b + 2*b^2)/(a*b))) + 1/4*sqrt((a*b*sqrt( -(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a ^3*b^3)) - 2*a^2 + 4*a*b - 2*b^2)/(a*b))*log((a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5)*x + (a^3*b^2*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a^3*b^3)) + a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*sqrt((a*b*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3* b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)/(a^3*b^3)) - 2*a^2 + 4*a*b - 2*b^2)/(a*b ))) - 1/4*sqrt((a*b*sqrt(-(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15...
Timed out. \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((a*x**4+b)/(x**2-1)/(b*x**4+a),x)
Output:
Timed out
Time = 0.10 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=\frac {1}{8} \, {\left (a - b\right )} {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {a} + \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} {\left (\sqrt {a} + \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} {\left (\sqrt {a} - \sqrt {b}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (\sqrt {a} - \sqrt {b}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}}\right )} - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \] Input:
integrate((a*x^4+b)/(x^2-1)/(b*x^4+a),x, algorithm="maxima")
Output:
1/8*(a - b)*(2*sqrt(2)*(sqrt(a) + sqrt(b))*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*s qrt(b))*sqrt(b)) + 2*sqrt(2)*(sqrt(a) + sqrt(b))*arctan(1/2*sqrt(2)*(2*sqr t(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqr t(a)*sqrt(b))*sqrt(b)) - sqrt(2)*(sqrt(a) - sqrt(b))*log(sqrt(b)*x^2 + sqr t(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + sqrt(2)*(sqrt(a) - s qrt(b))*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^ (3/4))) - 1/2*log(x + 1) + 1/2*log(x - 1)
Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.16 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (a - b\right )} + \left (a b^{3}\right )^{\frac {1}{4}} {\left (a b^{2} - b^{3}\right )}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (a - b\right )} + \left (a b^{3}\right )^{\frac {1}{4}} {\left (a b^{2} - b^{3}\right )}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (a - b\right )} - \left (a b^{3}\right )^{\frac {1}{4}} {\left (a b^{2} - b^{3}\right )}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} {\left (a - b\right )} - \left (a b^{3}\right )^{\frac {1}{4}} {\left (a b^{2} - b^{3}\right )}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \] Input:
integrate((a*x^4+b)/(x^2-1)/(b*x^4+a),x, algorithm="giac")
Output:
1/4*sqrt(2)*((a*b^3)^(3/4)*(a - b) + (a*b^3)^(1/4)*(a*b^2 - b^3))*arctan(1 /2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^3) + 1/4*sqrt(2)* ((a*b^3)^(3/4)*(a - b) + (a*b^3)^(1/4)*(a*b^2 - b^3))*arctan(1/2*sqrt(2)*( 2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^3) - 1/8*sqrt(2)*((a*b^3)^(3/ 4)*(a - b) - (a*b^3)^(1/4)*(a*b^2 - b^3))*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^3) + 1/8*sqrt(2)*((a*b^3)^(3/4)*(a - b) - (a*b^3)^(1/4)* (a*b^2 - b^3))*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^3) - 1/2* log(abs(x + 1)) + 1/2*log(abs(x - 1))
Time = 1.35 (sec) , antiderivative size = 6231, normalized size of antiderivative = 24.25 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx=\text {Too large to display} \] Input:
int((b + a*x^4)/((x^2 - 1)*(a + b*x^4)),x)
Output:
- atanh(x) - atan((((-(a^3*(-a^3*b^3)^(1/2) - b^3*(-a^3*b^3)^(1/2) + 2*a^2 *b^4 - 4*a^3*b^3 + 2*a^4*b^2 + 3*a*b^2*(-a^3*b^3)^(1/2) - 3*a^2*b*(-a^3*b^ 3)^(1/2))/(16*a^3*b^3))^(1/2)*((x*(32*a*b^8 + 16*b^9 - 272*a^2*b^7 - 320*a ^3*b^6 + 112*a^4*b^5 + 32*a^5*b^4 - 112*a^6*b^3) + (-(a^3*(-a^3*b^3)^(1/2) - b^3*(-a^3*b^3)^(1/2) + 2*a^2*b^4 - 4*a^3*b^3 + 2*a^4*b^2 + 3*a*b^2*(-a^ 3*b^3)^(1/2) - 3*a^2*b*(-a^3*b^3)^(1/2))/(16*a^3*b^3))^(1/2)*(128*a^2*b^7 - 64*a*b^8 + 512*a^3*b^6 + 384*a^4*b^5 + 64*a^5*b^4 + x*(-(a^3*(-a^3*b^3)^ (1/2) - b^3*(-a^3*b^3)^(1/2) + 2*a^2*b^4 - 4*a^3*b^3 + 2*a^4*b^2 + 3*a*b^2 *(-a^3*b^3)^(1/2) - 3*a^2*b*(-a^3*b^3)^(1/2))/(16*a^3*b^3))^(1/2)*(512*a^2 *b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)))*(-(a^3*(-a^3*b^3)^(1/2) - b^3*(-a^3*b^3)^(1/2) + 2*a^2*b^4 - 4*a^3*b^3 + 2*a^4*b^2 + 3*a*b^2*(-a^3 *b^3)^(1/2) - 3*a^2*b*(-a^3*b^3)^(1/2))/(16*a^3*b^3))^(1/2) + 20*a*b^8 + 4 *b^9 + 20*a^2*b^7 - 76*a^3*b^6 - 84*a^4*b^5 + 60*a^5*b^4 + 60*a^6*b^3 - 4* a^7*b^2) - x*(8*a*b^8 + 2*a^8*b + 6*b^9 - 12*a^2*b^7 - 16*a^3*b^6 + 8*a^4* b^5 + 8*a^5*b^4 - 4*a^6*b^3))*(-(a^3*(-a^3*b^3)^(1/2) - b^3*(-a^3*b^3)^(1/ 2) + 2*a^2*b^4 - 4*a^3*b^3 + 2*a^4*b^2 + 3*a*b^2*(-a^3*b^3)^(1/2) - 3*a^2* b*(-a^3*b^3)^(1/2))/(16*a^3*b^3))^(1/2)*1i - ((-(a^3*(-a^3*b^3)^(1/2) - b^ 3*(-a^3*b^3)^(1/2) + 2*a^2*b^4 - 4*a^3*b^3 + 2*a^4*b^2 + 3*a*b^2*(-a^3*b^3 )^(1/2) - 3*a^2*b*(-a^3*b^3)^(1/2))/(16*a^3*b^3))^(1/2)*(20*a*b^8 - (x*(32 *a*b^8 + 16*b^9 - 272*a^2*b^7 - 320*a^3*b^6 + 112*a^4*b^5 + 32*a^5*b^4 ...
Time = 0.17 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.18 \[ \int \frac {b+a x^4}{\left (-1+x^2\right ) \left (a+b x^4\right )} \, dx =\text {Too large to display} \] Input:
int((a*x^4+b)/(x^2-1)/(b*x^4+a),x)
Output:
( - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b )*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a + 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b* *(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b - 2* b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/( b**(1/4)*a**(1/4)*sqrt(2)))*a + 2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4) *a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b + 2*b**(1/ 4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/ 4)*a**(1/4)*sqrt(2)))*a - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1 /4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b + 2*b**(3/4)*a** (1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a** (1/4)*sqrt(2)))*a - 2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b + b**(1/4)*a**(3/4)*sq rt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*a - b** (1/4)*a**(3/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt (b)*x**2)*b - b**(1/4)*a**(3/4)*sqrt(2)*log(b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*a + b**(1/4)*a**(3/4)*sqrt(2)*log(b**(1/4)*a**(1/4 )*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*b - b**(3/4)*a**(1/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*a + b**(3/4)*a**(1/ 4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(b)*x**2)*b + b**(3/4)*a**(1/4)*sqrt(2)*log(b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) +...