Integrand size = 32, antiderivative size = 153 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\frac {2 \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \text {arctanh}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}} \] Output:
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) ^(1/2)/(3*d^2-2*(-64*a*e^3+d^4)^(1/2))^(1/2)-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)^(1/2)/(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 = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.46 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=-\text {RootSum}\left [8 a e^2-d^3 \text {$\#$1}+8 d e^2 \text {$\#$1}^3+8 e^3 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{d^3-24 d e^2 \text {$\#$1}^2-32 e^3 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-1),x]
Output:
-RootSum[8*a*e^2 - d^3*#1 + 8*d*e^2*#1^3 + 8*e^3*#1^4 & , Log[x - #1]/(d^3 - 24*d*e^2*#1^2 - 32*e^3*#1^3) & ]
Time = 0.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2458, 1406, 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.
| \(\displaystyle \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4}d\left (\frac {d}{4 e}+x\right )\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \frac {4 e^2 \int \frac {1}{8 e^3 \left (\frac {d}{4 e}+x\right )^2-\frac {1}{2} e \left (3 d^2+2 \sqrt {d^4-64 a e^3}\right )}d\left (\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}-\frac {4 e^2 \int \frac {1}{8 e^3 \left (\frac {d}{4 e}+x\right )^2-\frac {1}{2} e \left (3 d^2-2 \sqrt {d^4-64 a e^3}\right )}d\left (\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \text {arctanh}\left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \text {arctanh}\left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\) |
Input:
Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-1),x]
Output:
(2*ArcTanh[(4*e*(d/(4*e) + x))/Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqr t[d^4 - 64*a*e^3]*Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]) - (2*ArcTanh[(4*e* (d/(4*e) + x))/Sqrt[3*d^2 + 2*Sqrt[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]])
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)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 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 = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.44
| method | result | size |
| default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\) | \(67\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\) | \(67\) |
Input:
int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x,method=_RETURNVERBOSE)
Output:
sum(1/(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. 1115 vs. \(2 (133) = 266\).
Time = 0.09 (sec) , antiderivative size = 1115, normalized size of antiderivative = 7.29 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="fricas")
Output:
-sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960 *a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x + 2*(2*d^4 - 128*a*e^3 - 3*(5*d^10 - 64*a*d^6*e ^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/ sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^ 8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d) + sqrt((3*d^2 + 2*(5*d^8 - 64*a* d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x - 2* (2*d^4 - 128*a*e^3 - 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2*d^2*e^6)/sqrt(25 *d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^ 6)) + 2*d) - sqrt((3*d^2 - 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(2 5*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64 *a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x + 2*(2*d^4 - 128*a*e^3 + 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a ^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 - 2*(5*d^8 - 64*a*d^4*e^3 - 163 84*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3 *e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d) + sqrt((3*d^2 - 2...
Time = 1.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (1048576 a^{3} e^{9} - 12288 a^{2} d^{4} e^{6} - 384 a d^{8} e^{3} + 5 d^{12}\right ) + t^{2} \cdot \left (384 a d^{2} e^{3} - 6 d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (x + \frac {- 49152 t^{3} a^{2} d^{2} e^{6} - 192 t^{3} a d^{6} e^{3} + 15 t^{3} d^{10} + 256 t a e^{3} - 13 t d^{4} + 2 d}{8 e} \right )} \right )\right )} \] Input:
integrate(1/(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2),x)
Output:
RootSum(_t**4*(1048576*a**3*e**9 - 12288*a**2*d**4*e**6 - 384*a*d**8*e**3 + 5*d**12) + _t**2*(384*a*d**2*e**3 - 6*d**6) + 1, Lambda(_t, _t*log(x + ( -49152*_t**3*a**2*d**2*e**6 - 192*_t**3*a*d**6*e**3 + 15*_t**3*d**10 + 256 *_t*a*e**3 - 13*_t*d**4 + 2*d)/(8*e))))
\[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \frac {1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \] Input:
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="maxima")
Output:
integrate(1/(8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (133) = 266\).
Time = 0.12 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.77 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=-\frac {2 \, \log \left (x + \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{3} - 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{2} + 2 \, d^{3}} + \frac {2 \, \log \left (x - \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{3} + 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{2} - 2 \, d^{3}} - \frac {2 \, \log \left (x + \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{3} - 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{e}\right )}^{2} + 2 \, d^{3}} + \frac {2 \, \log \left (x - \frac {1}{4} \, \sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} + \frac {d}{4 \, e}\right )}{e^{3} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{3} + 3 \, d e^{2} {\left (\sqrt {\frac {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}}{e^{4}}} - \frac {d}{e}\right )}^{2} - 2 \, d^{3}} \] Input:
integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="giac")
Output:
-2*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) + 2*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) - 2*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) + 2*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)
Time = 23.15 (sec) , antiderivative size = 1264, normalized size of antiderivative = 8.26 \[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx =\text {Too large to display} \] Input:
int(1/(8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3),x)
Output:
atan((d^3*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2 )*3i - d^9*2i + a*d^5*e^3*256i - a^2*d*e^6*8192i - a^2*e^7*x*32768i - d^8* e*x*8i + d^2*e*x*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^ 6)^(1/2)*12i + a*d^4*e^4*x*1024i)/(5*d^12*(-(2*(d^12 - 262144*a^3*e^9 - 19 2*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2) + 1048576*a^3* e^9*(-(2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 1228 8*a^2*d^4*e^6))^(1/2) - 384*a*d^8*e^3*(-(2*(d^12 - 262144*a^3*e^9 - 192*a* d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048 576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2) - 12288*a^2*d^4*e^ 6*(-(2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288* a^2*d^4*e^6))^(1/2)))*(-(2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288* a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 38 4*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2)*2i - atan((d^3*(d^12 - 262144*a^3* e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2)*3i + d^9*2i - a*d^5*e^3*256 i + a^2*d*e^6*8192i + a^2*e^7*x*32768i + d^8*e*x*8i + d^2*e*x*(d^12 - 2621 44*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2)*12i - a*d^4*e^4*x*10 24i)/(5*d^12*((2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4...
\[ \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int \frac {1}{8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}d x \] Input:
int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x)
Output:
int(1/(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4),x)