Integrand size = 32, antiderivative size = 332 \[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\frac {(d+4 e x) \left (13 d^4-256 a e^3-3 d^2 (d+4 e x)^2\right )}{2 \left (5 d^8-64 a d^4 e^3-16384 a^2 e^6\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}-\frac {24 e \left (d^4+128 a e^3-d^2 \sqrt {d^4-64 a e^3}\right ) \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (5 d^4+256 a e^3\right ) \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}+\frac {24 e \left (d^4+128 a e^3+d^2 \sqrt {d^4-64 a e^3}\right ) \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (5 d^4+256 a e^3\right ) \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}} \] Output:
1/2*(4*e*x+d)*(13*d^4-256*a*e^3-3*d^2*(4*e*x+d)^2)/(-16384*a^2*e^6-64*a*d^ 4*e^3+5*d^8)/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)-24*e*(d^4+128*a*e^3-d^2 *(-64*a*e^3+d^4)^(1/2))*arctanh((4*e*x+d)/(3*d^2-2*(-64*a*e^3+d^4)^(1/2))^ (1/2))/(-64*a*e^3+d^4)^(3/2)/(256*a*e^3+5*d^4)/(3*d^2-2*(-64*a*e^3+d^4)^(1 /2))^(1/2)+24*e*(d^4+128*a*e^3+d^2*(-64*a*e^3+d^4)^(1/2))*arctanh((4*e*x+d )/(3*d^2+2*(-64*a*e^3+d^4)^(1/2))^(1/2))/(-64*a*e^3+d^4)^(3/2)/(256*a*e^3+ 5*d^4)/(3*d^2+2*(-64*a*e^3+d^4)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\frac {(d+4 e x) \left (5 d^4-128 a e^3-12 d^3 e x-24 d^2 e^2 x^2\right )}{\left (d^4-64 a e^3\right ) \left (5 d^4+256 a e^3\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}+\frac {48 e^2 \text {RootSum}\left [8 a e^2-d^3 \text {$\#$1}+8 d e^2 \text {$\#$1}^3+8 e^3 \text {$\#$1}^4\&,\frac {32 a e^2 \log (x-\text {$\#$1})+d^3 \log (x-\text {$\#$1}) \text {$\#$1}+2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^2}{-d^3+24 d e^2 \text {$\#$1}^2+32 e^3 \text {$\#$1}^3}\&\right ]}{-5 d^8+64 a d^4 e^3+16384 a^2 e^6} \] Input:
Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-2),x]
Output:
((d + 4*e*x)*(5*d^4 - 128*a*e^3 - 12*d^3*e*x - 24*d^2*e^2*x^2))/((d^4 - 64 *a*e^3)*(5*d^4 + 256*a*e^3)*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)) + (48*e^2*RootSum[8*a*e^2 - d^3*#1 + 8*d*e^2*#1^3 + 8*e^3*#1^4 & , (32*a*e^ 2*Log[x - #1] + d^3*Log[x - #1]*#1 + 2*d^2*e*Log[x - #1]*#1^2)/(-d^3 + 24* d*e^2*#1^2 + 32*e^3*#1^3) & ])/(-5*d^8 + 64*a*d^4*e^3 + 16384*a^2*e^6)
Time = 0.57 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2458, 1405, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\left (\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^2}d\left (\frac {d}{4 e}+x\right )\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {64 e \left (\frac {d}{4 e}+x\right ) \left (-256 a e^3+13 d^4-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )}-\frac {4 \int -\frac {48 e^2 \left (d^4-16 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2-256 a e^3\right )}{\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}d\left (\frac {d}{4 e}+x\right )}{e \left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {192 e \int \frac {d^4-16 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2-256 a e^3}{\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}d\left (\frac {d}{4 e}+x\right )}{-16384 a^2 e^6-64 a d^4 e^3+5 d^8}+\frac {64 e \left (\frac {d}{4 e}+x\right ) \left (-256 a e^3+13 d^4-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {192 e \left (\frac {8 e^2 \left (-d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \int \frac {1}{256 e^3 \left (\frac {d}{4 e}+x\right )^2-16 e \left (3 d^2-2 \sqrt {d^4-64 a e^3}\right )}d\left (\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}-\frac {8 e^2 \left (d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \int \frac {1}{256 e^3 \left (\frac {d}{4 e}+x\right )^2-16 e \left (3 d^2+2 \sqrt {d^4-64 a e^3}\right )}d\left (\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}\right )}{-16384 a^2 e^6-64 a d^4 e^3+5 d^8}+\frac {64 e \left (\frac {d}{4 e}+x\right ) \left (-256 a e^3+13 d^4-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {192 e \left (\frac {\left (d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \text {arctanh}\left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{8 \sqrt {d^4-64 a e^3} \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}-\frac {\left (-d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \text {arctanh}\left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{8 \sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{-16384 a^2 e^6-64 a d^4 e^3+5 d^8}+\frac {64 e \left (\frac {d}{4 e}+x\right ) \left (-256 a e^3+13 d^4-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )}\) |
Input:
Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-2),x]
Output:
(64*e*(d/(4*e) + x)*(13*d^4 - 256*a*e^3 - 48*d^2*e^2*(d/(4*e) + x)^2))/((5 *d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)*((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/ (4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4)) + (192*e*(-1/8*((d^4 + 128*a*e^3 - d^2*Sqrt[d^4 - 64*a*e^3])*ArcTanh[(4*e*(d/(4*e) + x))/Sqrt[3*d^2 - 2*Sqr t[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a* e^3]]) + ((d^4 + 128*a*e^3 + d^2*Sqrt[d^4 - 64*a*e^3])*ArcTanh[(4*e*(d/(4* e) + x))/Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]])/(8*Sqrt[d^4 - 64*a*e^3]*Sq rt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^ 6)
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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {\frac {12 d^{2} e^{3} x^{3}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}+\frac {9 d^{3} e^{2} x^{2}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}+\frac {e x}{256 e^{3} a +5 d^{4}}+\frac {d \left (128 e^{3} a -5 d^{4}\right )}{131072 a^{2} e^{6}+512 a \,d^{4} e^{3}-40 d^{8}}}{x^{4} e^{3}+d \,e^{2} x^{3}-\frac {1}{8} d^{3} x +a \,e^{2}}+\frac {384 e^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\left (2 \textit {\_R}^{2} d^{2} e +\textit {\_R} \,d^{3}+32 a \,e^{2}\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\right )}{\left (2048 e^{3} a +40 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}\) | \(288\) |
| risch | \(\frac {\frac {12 d^{2} e^{3} x^{3}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}+\frac {9 d^{3} e^{2} x^{2}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}+\frac {e x}{256 e^{3} a +5 d^{4}}+\frac {d \left (128 e^{3} a -5 d^{4}\right )}{131072 a^{2} e^{6}+512 a \,d^{4} e^{3}-40 d^{8}}}{x^{4} e^{3}+d \,e^{2} x^{3}-\frac {1}{8} d^{3} x +a \,e^{2}}+48 e^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\left (\frac {2 d^{2} e \,\textit {\_R}^{2}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}+\frac {d^{3} \textit {\_R}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}+\frac {32 a \,e^{2}}{\left (256 e^{3} a +5 d^{4}\right ) \left (64 e^{3} a -d^{4}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\right )\) | \(344\) |
Input:
int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x,method=_RETURNVERBOSE)
Output:
(12*d^2*e^3/(256*a*e^3+5*d^4)/(64*a*e^3-d^4)*x^3+9*d^3*e^2/(256*a*e^3+5*d^ 4)/(64*a*e^3-d^4)*x^2+e/(256*a*e^3+5*d^4)*x+1/8*d*(128*a*e^3-5*d^4)/(16384 *a^2*e^6+64*a*d^4*e^3-5*d^8))/(x^4*e^3+d*e^2*x^3-1/8*d^3*x+a*e^2)+384*e^2/ (2048*a*e^3+40*d^4)/(64*a*e^3-d^4)*sum((2*_R^2*d^2*e+_R*d^3+32*a*e^2)/(32* _R^3*e^3+24*_R^2*d*e^2-d^3)*ln(x-_R),_R=RootOf(8*_Z^4*e^3+8*_Z^3*d*e^2-_Z* d^3+8*a*e^2))
Leaf count of result is larger than twice the leaf count of optimal. 4285 vs. \(2 (306) = 612\).
Time = 0.27 (sec) , antiderivative size = 4285, normalized size of antiderivative = 12.91 \[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**2,x)
Output:
Timed out
\[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}\right )}^{2}} \,d x } \] Input:
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="maxima")
Output:
-48*e^2*integrate((2*d^2*e*x^2 + d^3*x + 32*a*e^2)/(8*e^3*x^4 + 8*d*e^2*x^ 3 - d^3*x + 8*a*e^2), x)/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6) - (96*d^2* e^3*x^3 + 72*d^3*e^2*x^2 - 5*d^5 + 128*a*d*e^3 - 8*(d^4*e - 64*a*e^4)*x)/( 40*a*d^8*e^2 - 512*a^2*d^4*e^5 - 131072*a^3*e^8 + 8*(5*d^8*e^3 - 64*a*d^4* e^6 - 16384*a^2*e^9)*x^4 + 8*(5*d^9*e^2 - 64*a*d^5*e^5 - 16384*a^2*d*e^8)* x^3 - (5*d^11 - 64*a*d^7*e^3 - 16384*a^2*d^3*e^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (306) = 612\).
Time = 0.12 (sec) , antiderivative size = 1115, normalized size of antiderivative = 3.36 \[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="giac")
Output:
12*((d^2*e^3*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) + d/e)^2 - 2*d^3*e^2*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) + d/e) + 2 56*a*e^4)*log(x + 1/4*sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) + 1/4*d/e)/(e^3*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) + d/e)^ 3 - 3*d*e^2*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) + d/e)^2 + 2*d^3) - (d^2*e^3*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) - d /e)^2 + 2*d^3*e^2*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) - d/ e) + 256*a*e^4)*log(x - 1/4*sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/ e^4) + 1/4*d/e)/(e^3*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) - d/e)^3 + 3*d*e^2*(sqrt((3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4) - d/ e)^2 - 2*d^3) + (d^2*e^3*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^ 4) + d/e)^2 - 2*d^3*e^2*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4 ) + d/e) + 256*a*e^4)*log(x + 1/4*sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3) *e^2)/e^4) + 1/4*d/e)/(e^3*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)/ e^4) + d/e)^3 - 3*d*e^2*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)/e^4 ) + d/e)^2 + 2*d^3) - (d^2*e^3*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e ^2)/e^4) - d/e)^2 + 2*d^3*e^2*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^ 2)/e^4) - d/e) + 256*a*e^4)*log(x - 1/4*sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64* a*e^3)*e^2)/e^4) + 1/4*d/e)/(e^3*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3) *e^2)/e^4) - d/e)^3 + 3*d*e^2*(sqrt((3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)...
Time = 26.79 (sec) , antiderivative size = 10351, normalized size of antiderivative = 31.18 \[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^2,x)
Output:
((8*e*x)/(256*a*e^3 + 5*d^4) - (5*d^5 - 128*a*d*e^3)/((64*a*e^3 - d^4)*(25 6*a*e^3 + 5*d^4)) + (72*d^3*e^2*x^2)/((64*a*e^3 - d^4)*(256*a*e^3 + 5*d^4) ) + (96*d^2*e^3*x^3)/((64*a*e^3 - d^4)*(256*a*e^3 + 5*d^4)))/(8*a*e^2 - d^ 3*x + 8*e^3*x^4 + 8*d*e^2*x^3) + atan((((288*(d^22*e^2 + d^4*e^2*(-(64*a*e ^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10 *e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(6 4*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800 *a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344* a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*((( 1536*(68719476736*a^5*e^24 + 20*d^20*e^9 - 7936*a*d^16*e^12 + 770048*a^2*d ^12*e^15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 171 79869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d ^8*e^9 + 134217728*a^4*d^4*e^12) - ((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^20 - 25769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d^20 - 171798 69184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8* e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^12 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^4*d^6*e^21 - 1 7179869184*a^5*d^2*e^24))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^...
\[ \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx=\int \frac {1}{64 e^{6} x^{8}+128 d \,e^{5} x^{7}+64 d^{2} e^{4} x^{6}-16 d^{3} e^{3} x^{5}+128 a \,e^{5} x^{4}-16 d^{4} e^{2} x^{4}+128 a d \,e^{4} x^{3}+d^{6} x^{2}-16 a \,d^{3} e^{2} x +64 a^{2} e^{4}}d x \] Input:
int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x)
Output:
int(1/(64*a**2*e**4 - 16*a*d**3*e**2*x + 128*a*d*e**4*x**3 + 128*a*e**5*x* *4 + d**6*x**2 - 16*d**4*e**2*x**4 - 16*d**3*e**3*x**5 + 64*d**2*e**4*x**6 + 128*d*e**5*x**7 + 64*e**6*x**8),x)