Integrand size = 22, antiderivative size = 113 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ] \] Output:
Unintegrable
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.26 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\frac {\sqrt [4]{x^2+x^4} \left (2 \sqrt [4]{2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )-\text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ]\right )}{8 \sqrt {x} \sqrt [4]{1+x^2}} \] Input:
Integrate[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]
Output:
((x^2 + x^4)^(1/4)*(2*2^(1/4)*(ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)] - ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)]) - RootSum[2 - 2*#1^4 + #1^8 & , (-(Log[Sqrt[x]]*#1) + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1]*#1)/(-1 + #1^4) & ]))/(8*Sqrt[x]*(1 + x^2)^(1/4))
Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2467, 25, 2019, 2035, 2461, 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.
\(\displaystyle \int \frac {x^4 \sqrt [4]{x^4+x^2}}{x^8-1} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x^4+x^2} \int -\frac {x^{9/2} \sqrt [4]{x^2+1}}{1-x^8}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x^4+x^2} \int \frac {x^{9/2} \sqrt [4]{x^2+1}}{1-x^8}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle -\frac {\sqrt [4]{x^4+x^2} \int \frac {x^{9/2}}{\left (x^2+1\right )^{3/4} \left (-x^6+x^4-x^2+1\right )}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \int \frac {x^5}{\left (x^2+1\right )^{3/4} \left (-x^6+x^4-x^2+1\right )}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
\(\Big \downarrow \) 2461 |
\(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \int \left (-\frac {x^5}{4 (x-1) \left (x^2+1\right )^{3/4}}+\frac {x^5}{4 (x+1) \left (x^2+1\right )^{3/4}}+\frac {\sqrt [4]{x^2+1} x^5}{2 \left (x^4+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \left (-\frac {1}{12} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-i x^2,-x^2\right )-\frac {1}{12} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},i x^2,-x^2\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{4\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{4\ 2^{3/4}}\right )}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
Input:
Int[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]
Output:
(-2*(x^2 + x^4)^(1/4)*(-1/12*(x^(3/2)*AppellF1[3/4, 1, -1/4, 7/4, (-I)*x^2 , -x^2]) - (x^(3/2)*AppellF1[3/4, 1, -1/4, 7/4, I*x^2, -x^2])/12 - ArcTan[ (2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)]/(4*2^(3/4)) + ArcTanh[(2^(1/4)*Sqrt[x]) /(1 + x^2)^(1/4)]/(4*2^(3/4))))/(Sqrt[x]*(1 + x^2)^(1/4))
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 194.84 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-1}\right )}{8}-\frac {2^{\frac {1}{4}} \ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right )}{8}-\frac {2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right )}{4}\) | \(117\) |
trager | \(\text {Expression too large to display}\) | \(4795\) |
Input:
int(x^4*(x^4+x^2)^(1/4)/(x^8-1),x,method=_RETURNVERBOSE)
Output:
-1/8*sum(_R*ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(_R^4-1),_R=RootOf(_Z^8-2*_Z ^4+2))-1/8*2^(1/4)*ln((2^(1/4)*x+(x^2*(x^2+1))^(1/4))/(-2^(1/4)*x+(x^2*(x^ 2+1))^(1/4)))-1/4*2^(1/4)*arctan(1/2*2^(3/4)/x*(x^2*(x^2+1))^(1/4))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.40 (sec) , antiderivative size = 1047, normalized size of antiderivative = 9.27 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(x^4+x^2)^(1/4)/(x^8-1),x, algorithm="fricas")
Output:
1/32*8^(3/4)*arctan(1/8*(8^(3/4)*(x^4 + x^2)^(3/4) - 2*8^(1/4)*(x^4 + x^2) ^(1/4)*(x^2 + 1))/(x^3 + x)) - 1/64*8^(3/4)*log((4*sqrt(2)*(x^4 + x^2)^(1/ 4)*x^2 + 8^(3/4)*sqrt(x^4 + x^2)*x + 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^( 3/4))/(x^3 - x)) + 1/64*8^(3/4)*log((4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 - 8^( 3/4)*sqrt(x^4 + x^2)*x - 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) + 1/16*sqrt(-sqrt(I + 1))*log(-(4*sqrt(I + 1)*((3*I + 4)*x^4 - (4*I - 3)*x^2)*(x^4 + x^2)^(1/4) + 4*(x^4 + x^2)^(3/4)*((3*I + 4)*x^2 - 4*I + 3) - (sqrt(I + 1)*(-(15*I - 5)*x^5 - (12*I + 16)*x^3 + (I - 7)*x) - 4*sqrt(x ^4 + x^2)*((4*I - 3)*x^3 + (3*I + 4)*x))*sqrt(-sqrt(I + 1)))/(x^5 + x)) - 1/16*sqrt(-sqrt(I + 1))*log(-(4*sqrt(I + 1)*((3*I + 4)*x^4 - (4*I - 3)*x^2 )*(x^4 + x^2)^(1/4) + 4*(x^4 + x^2)^(3/4)*((3*I + 4)*x^2 - 4*I + 3) - (sqr t(I + 1)*((15*I - 5)*x^5 + (12*I + 16)*x^3 - (I - 7)*x) - 4*sqrt(x^4 + x^2 )*(-(4*I - 3)*x^3 - (3*I + 4)*x))*sqrt(-sqrt(I + 1)))/(x^5 + x)) + 1/16*sq rt(-sqrt(-I + 1))*log(-(4*sqrt(-I + 1)*(-(3*I - 4)*x^4 + (4*I + 3)*x^2)*(x ^4 + x^2)^(1/4) + 4*(x^4 + x^2)^(3/4)*(-(3*I - 4)*x^2 + 4*I + 3) - (sqrt(- I + 1)*((15*I + 5)*x^5 + (12*I - 16)*x^3 - (I + 7)*x) - 4*sqrt(x^4 + x^2)* (-(4*I + 3)*x^3 - (3*I - 4)*x))*sqrt(-sqrt(-I + 1)))/(x^5 + x)) - 1/16*sqr t(-sqrt(-I + 1))*log(-(4*sqrt(-I + 1)*(-(3*I - 4)*x^4 + (4*I + 3)*x^2)*(x^ 4 + x^2)^(1/4) + 4*(x^4 + x^2)^(3/4)*(-(3*I - 4)*x^2 + 4*I + 3) - (sqrt(-I + 1)*(-(15*I + 5)*x^5 - (12*I - 16)*x^3 + (I + 7)*x) - 4*sqrt(x^4 + x^...
Not integrable
Time = 0.72 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.28 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int \frac {x^{4} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \] Input:
integrate(x**4*(x**4+x**2)**(1/4)/(x**8-1),x)
Output:
Integral(x**4*(x**2*(x**2 + 1))**(1/4)/((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)), x)
Not integrable
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.47 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int { \frac {{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{4}}{x^{8} - 1} \,d x } \] Input:
integrate(x^4*(x^4+x^2)^(1/4)/(x^8-1),x, algorithm="maxima")
Output:
-2/5*(x^3 + x)*(x^2 + 1)^(1/4)*x^(9/2)/(x^8 - 1) - integrate(16/5*(x^2 + 1 )^(5/4)*x^(9/2)/(x^16 - 2*x^8 + 1), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.93 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx =\text {Too large to display} \] Input:
integrate(x^4*(x^4+x^2)^(1/4)/(x^8-1),x, algorithm="giac")
Output:
-1/4*2^(1/4)*arctan(1/2*2^(3/4)*(1/x^2 + 1)^(1/4)) - 8*I*(-1/16777216*I + 1/16777216)^(1/4)*log((-4722366482869645213696*I + 4722366482869645213696) ^(1/4) - 262144*(1/x^2 + 1)^(1/4)) + 8*I*(-1/16777216*I + 1/16777216)^(1/4 )*log(-(-4722366482869645213696*I + 4722366482869645213696)^(1/4) - 262144 *(1/x^2 + 1)^(1/4)) + I*(1/4096*I + 1/4096)^(1/4)*log((68719476736*I + 687 19476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) - (1/4096*I + 1/4096)^(1/4)*log(I *(68719476736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) + (1/4096*I + 1/4096)^(1/4)*log(-I*(68719476736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) - I*(1/4096*I + 1/4096)^(1/4)*log(-(68719476736*I + 68719476736) ^(1/4) - 512*(1/x^2 + 1)^(1/4)) + (-1/4096*I + 1/4096)^(1/4)*log(I*(-68719 476736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) - (-1/4096*I + 1/40 96)^(1/4)*log(-I*(-68719476736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1 /4)) - 1/8*2^(1/4)*log(2^(1/4) + (1/x^2 + 1)^(1/4)) + 1/8*2^(1/4)*log(abs( -2^(1/4) + (1/x^2 + 1)^(1/4)))
Not integrable
Time = 7.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int \frac {x^4\,{\left (x^4+x^2\right )}^{1/4}}{x^8-1} \,d x \] Input:
int((x^4*(x^2 + x^4)^(1/4))/(x^8 - 1),x)
Output:
int((x^4*(x^2 + x^4)^(1/4))/(x^8 - 1), x)
Not integrable
Time = 0.67 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {1}{4}} x^{4}}{x^{8}-1}d x \] Input:
int(x^4*(x^4+x^2)^(1/4)/(x^8-1),x)
Output:
int((sqrt(x)*(x**2 + 1)**(1/4)*x**4)/(x**8 - 1),x)