Integrand size = 19, antiderivative size = 129 \[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx=-\frac {x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right ),\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{8 \sqrt {3} \sqrt [4]{29} \sqrt {8+8 x-x^3+8 x^4}} \]
-1/696*x^2*(cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))^2)^(1/2)/cos(2*ar ctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))*EllipticF(sin(2*arctan(1/87*(4+x)*29^ (3/4)/x*3^(1/2))),1/58*(1682+58*29^(1/2))^(1/2))*(87+(4+x)^2*29^(1/2)/x^2) *((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*29^(1/2)/x^2)^2)^(1/2)*29^(3/4)* 3^(1/2)/(8*x^4-x^3+8*x+8)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 10.70 (sec) , antiderivative size = 927, normalized size of antiderivative = 7.19 \[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx =\text {Too large to display} \]
(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*( Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], ((R oot[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8* #1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - R oot[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(x - Root[8 + 8*#1 - #1^3 + 8*#1 ^4 & , 2, 0])^2*Sqrt[((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])* Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8* #1 - #1^3 + 8*#1^4 & , 4, 0])^2)])/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[...
Time = 0.40 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.30, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2504, 27, 7270, 1416}
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.
\(\displaystyle \int \frac {1}{\sqrt {8 x^4-x^3+8 x+8}} \, dx\) |
\(\Big \downarrow \) 2504 |
\(\displaystyle -1024 \int \frac {1}{128 \sqrt {2} \left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}{\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^4}}}d\left (\frac {1}{4}+\frac {1}{x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 \sqrt {2} \int \frac {1}{\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}{\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^4}}}d\left (\frac {1}{4}+\frac {1}{x}\right )\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -\frac {4 \sqrt {2} \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \int \frac {1}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {\left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {3} \sqrt [4]{29}}\right ),\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{\sqrt {6} \sqrt [4]{29} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\) |
-(((87 + 16*Sqrt[29]*(1/4 + x^(-1))^2)*Sqrt[(261 - 96*(1/4 + x^(-1))^2 + 2 56*(1/4 + x^(-1))^4)/(87 + 16*Sqrt[29]*(1/4 + x^(-1))^2)^2]*EllipticF[2*Ar cTan[(4*(1/4 + x^(-1)))/(Sqrt[3]*29^(1/4))], (29 + Sqrt[29])/58])/(Sqrt[6] *29^(1/4)*(1 - 4*(1/4 + x^(-1)))^2*Sqrt[(261 - 96*(1/4 + x^(-1))^2 + 256*( 1/4 + x^(-1))^4)/(1 - 4*(1/4 + x^(-1)))^4]))
3.8.96.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.13 (sec) , antiderivative size = 965, normalized size of antiderivative = 7.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(965\) |
elliptic | \(\text {Expression too large to display}\) | \(965\) |
1/2*(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4) )*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)) *(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4 )-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2 )))^(1/2)*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))^2*((RootOf(8*_Z^4-_Z^3+8* _Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8* _Z+8,index=3))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)-RootOf(8*_Z^4-_Z^3+8*_Z +8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*((RootOf(8*_Z^4 -_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4 -_Z^3+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_ Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)/(RootOf (8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/(RootOf(8 *_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*2^(1/2)/((x -RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2) )*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,inde x=4)))^(1/2)*EllipticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4- _Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^ 4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^ 4-_Z^3+8*_Z+8,index=2)))^(1/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf (8*_Z^4-_Z^3+8*_Z+8,index=3))*(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)-RootO...
\[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx=\int { \frac {1}{\sqrt {8 \, x^{4} - x^{3} + 8 \, x + 8}} \,d x } \]
\[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx=\int \frac {1}{\sqrt {8 x^{4} - x^{3} + 8 x + 8}}\, dx \]
\[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx=\int { \frac {1}{\sqrt {8 \, x^{4} - x^{3} + 8 \, x + 8}} \,d x } \]
\[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx=\int { \frac {1}{\sqrt {8 \, x^{4} - x^{3} + 8 \, x + 8}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {8+8 x-x^3+8 x^4}} \, dx=\int \frac {1}{\sqrt {8\,x^4-x^3+8\,x+8}} \,d x \]