Integrand size = 13, antiderivative size = 157 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \arctan \left (1+\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}} \]
-45/32/a^12/x+1/8/a^4/x/(a^4+x^4)^2+9/32/a^8/x/(a^4+x^4)+45/128*arctan(1-x *2^(1/2)/a)/a^13*2^(1/2)-45/128*arctan(1+x*2^(1/2)/a)/a^13*2^(1/2)-45/256* ln(a^2+x^2-a*x*2^(1/2))/a^13*2^(1/2)+45/256*ln(a^2+x^2+a*x*2^(1/2))/a^13*2 ^(1/2)
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=-\frac {\frac {256 a}{x}+\frac {32 a^5 x^3}{\left (a^4+x^4\right )^2}+\frac {104 a x^3}{a^4+x^4}-90 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} x}{a}\right )+90 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} x}{a}\right )+45 \sqrt {2} \log \left (a^2-\sqrt {2} a x+x^2\right )-45 \sqrt {2} \log \left (a^2+\sqrt {2} a x+x^2\right )}{256 a^{13}} \]
-1/256*((256*a)/x + (32*a^5*x^3)/(a^4 + x^4)^2 + (104*a*x^3)/(a^4 + x^4) - 90*Sqrt[2]*ArcTan[1 - (Sqrt[2]*x)/a] + 90*Sqrt[2]*ArcTan[1 + (Sqrt[2]*x)/ a] + 45*Sqrt[2]*Log[a^2 - Sqrt[2]*a*x + x^2] - 45*Sqrt[2]*Log[a^2 + Sqrt[2 ]*a*x + x^2])/a^13
Time = 0.36 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {819, 819, 847, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {9 \int \frac {1}{x^2 \left (a^4+x^4\right )^2}dx}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{x^2 \left (a^4+x^4\right )}dx}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\int \frac {x^2}{a^4+x^4}dx}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \int \frac {a^2+x^2}{a^4+x^4}dx-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{a^2-\sqrt {2} x a+x^2}dx+\frac {1}{2} \int \frac {1}{a^2+\sqrt {2} x a+x^2}dx\right )-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} x}{a}\right )^2-1}d\left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} x}{a}+1\right )^2-1}d\left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}\right )-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} a-2 x}{a^2-\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}+\frac {\int -\frac {\sqrt {2} \left (a+\sqrt {2} x\right )}{a^2+\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} a-2 x}{a^2-\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}-\frac {\int \frac {\sqrt {2} \left (a+\sqrt {2} x\right )}{a^2+\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} a-2 x}{a^2-\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}-\frac {\int \frac {a+\sqrt {2} x}{a^2+\sqrt {2} x a+x^2}dx}{2 a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )}{a^4}-\frac {1}{a^4 x}\right )}{4 a^4}+\frac {1}{4 a^4 x \left (a^4+x^4\right )}\right )}{8 a^4}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9 \left (\frac {1}{4 a^4 x \left (a^4+x^4\right )}+\frac {5 \left (-\frac {1}{a^4 x}-\frac {\frac {1}{2} \left (\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{2 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{2 \sqrt {2} a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )}{a^4}\right )}{4 a^4}\right )}{8 a^4}\) |
1/(8*a^4*x*(a^4 + x^4)^2) + (9*(1/(4*a^4*x*(a^4 + x^4)) + (5*(-(1/(a^4*x)) - ((-(ArcTan[1 - (Sqrt[2]*x)/a]/(Sqrt[2]*a)) + ArcTan[1 + (Sqrt[2]*x)/a]/ (Sqrt[2]*a))/2 + (Log[a^2 - Sqrt[2]*a*x + x^2]/(2*Sqrt[2]*a) - Log[a^2 + S qrt[2]*a*x + x^2]/(2*Sqrt[2]*a))/2)/a^4))/(4*a^4)))/(8*a^4)
3.2.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {-\frac {45 x^{8}}{32 a^{12}}-\frac {81 x^{4}}{32 a^{8}}-\frac {1}{a^{4}}}{x \left (a^{4}+x^{4}\right )^{2}}+\frac {45 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{52} \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4} a^{52}+4\right ) x +\textit {\_R}^{3} a^{40}\right )\right )}{128}\) | \(75\) |
| default | \(-\frac {\frac {\frac {17}{32} a^{4} x^{3}+\frac {13}{32} x^{7}}{\left (a^{4}+x^{4}\right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (a^{4}\right )^{\frac {1}{4}}}}{a^{12}}-\frac {1}{a^{12} x}\) | \(124\) |
(-45/32/a^12*x^8-81/32/a^8*x^4-1/a^4)/x/(a^4+x^4)^2+45/128*sum(_R*ln((5*_R ^4*a^52+4)*x+_R^3*a^40),_R=RootOf(_Z^4*a^52+1))
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=-\frac {128 \, a^{8} + 324 \, a^{4} x^{4} + 180 \, x^{8} + 45 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \left (-\frac {1}{a^{52}}\right )^{\frac {1}{4}} \log \left (a^{40} \left (-\frac {1}{a^{52}}\right )^{\frac {3}{4}} + x\right ) + 45 \, {\left (-i \, a^{20} x - 2 i \, a^{16} x^{5} - i \, a^{12} x^{9}\right )} \left (-\frac {1}{a^{52}}\right )^{\frac {1}{4}} \log \left (i \, a^{40} \left (-\frac {1}{a^{52}}\right )^{\frac {3}{4}} + x\right ) + 45 \, {\left (i \, a^{20} x + 2 i \, a^{16} x^{5} + i \, a^{12} x^{9}\right )} \left (-\frac {1}{a^{52}}\right )^{\frac {1}{4}} \log \left (-i \, a^{40} \left (-\frac {1}{a^{52}}\right )^{\frac {3}{4}} + x\right ) - 45 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \left (-\frac {1}{a^{52}}\right )^{\frac {1}{4}} \log \left (-a^{40} \left (-\frac {1}{a^{52}}\right )^{\frac {3}{4}} + x\right )}{128 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )}} \]
-1/128*(128*a^8 + 324*a^4*x^4 + 180*x^8 + 45*(a^20*x + 2*a^16*x^5 + a^12*x ^9)*(-1/a^52)^(1/4)*log(a^40*(-1/a^52)^(3/4) + x) + 45*(-I*a^20*x - 2*I*a^ 16*x^5 - I*a^12*x^9)*(-1/a^52)^(1/4)*log(I*a^40*(-1/a^52)^(3/4) + x) + 45* (I*a^20*x + 2*I*a^16*x^5 + I*a^12*x^9)*(-1/a^52)^(1/4)*log(-I*a^40*(-1/a^5 2)^(3/4) + x) - 45*(a^20*x + 2*a^16*x^5 + a^12*x^9)*(-1/a^52)^(1/4)*log(-a ^40*(-1/a^52)^(3/4) + x))/(a^20*x + 2*a^16*x^5 + a^12*x^9)
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=\frac {- 32 a^{8} - 81 a^{4} x^{4} - 45 x^{8}}{32 a^{20} x + 64 a^{16} x^{5} + 32 a^{12} x^{9}} + \frac {\operatorname {RootSum} {\left (268435456 t^{4} + 4100625, \left ( t \mapsto t \log {\left (- \frac {2097152 t^{3} a}{91125} + x \right )} \right )\right )}}{a^{13}} \]
(-32*a**8 - 81*a**4*x**4 - 45*x**8)/(32*a**20*x + 64*a**16*x**5 + 32*a**12 *x**9) + RootSum(268435456*_t**4 + 4100625, Lambda(_t, _t*log(-2097152*_t* *3*a/91125 + x)))/a**13
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=-\frac {32 \, a^{8} + 81 \, a^{4} x^{4} + 45 \, x^{8}}{32 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )}} - \frac {45 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{a} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{a}\right )}}{256 \, a^{12}} \]
-1/32*(32*a^8 + 81*a^4*x^4 + 45*x^8)/(a^20*x + 2*a^16*x^5 + a^12*x^9) - 45 /256*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a + 2*x)/a)/a + 2*sqrt(2)*arct an(-1/2*sqrt(2)*(sqrt(2)*a - 2*x)/a)/a - sqrt(2)*log(sqrt(2)*a*x + a^2 + x ^2)/a + sqrt(2)*log(-sqrt(2)*a*x + a^2 + x^2)/a)/a^12
Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=-\frac {45 \, \sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{128 \, a^{14}} - \frac {45 \, \sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{128 \, a^{14}} + \frac {45 \, \sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{256 \, a^{14}} - \frac {45 \, \sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{256 \, a^{14}} - \frac {17 \, a^{4} x^{3} + 13 \, x^{7}}{32 \, {\left (a^{4} + x^{4}\right )}^{2} a^{12}} - \frac {1}{a^{12} x} \]
-45/128*sqrt(2)*abs(a)*arctan(1/2*sqrt(2)*(sqrt(2)*abs(a) + 2*x)/abs(a))/a ^14 - 45/128*sqrt(2)*abs(a)*arctan(-1/2*sqrt(2)*(sqrt(2)*abs(a) - 2*x)/abs (a))/a^14 + 45/256*sqrt(2)*abs(a)*log(sqrt(2)*x*abs(a) + x^2 + abs(a)^2)/a ^14 - 45/256*sqrt(2)*abs(a)*log(-sqrt(2)*x*abs(a) + x^2 + abs(a)^2)/a^14 - 1/32*(17*a^4*x^3 + 13*x^7)/((a^4 + x^4)^2*a^12) - 1/(a^12*x)
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx=\frac {45\,{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{64\,a^{13}}-\frac {45\,{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{64\,a^{13}}-\frac {\frac {1}{a^4}+\frac {81\,x^4}{32\,a^8}+\frac {45\,x^8}{32\,a^{12}}}{a^8\,x+2\,a^4\,x^5+x^9} \]