Integrand size = 22, antiderivative size = 113 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.26 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\frac {\sqrt [4]{x^2+x^4} \left (2 \sqrt [4]{2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )-\text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ]\right )}{8 \sqrt {x} \sqrt [4]{1+x^2}} \]
((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 = 1.01 (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 |
\(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \left (-\frac {1}{12} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-i x^2,-x^2\right )-\frac {1}{12} x^{3/2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},i x^2,-x^2\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{4\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{4\ 2^{3/4}}\right )}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
(-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))
3.17.91.3.1 Defintions of rubi rules used
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 = 45.01 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-1}\right )}{8}-\frac {2^{\frac {1}{4}} \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right )}{8}-\frac {2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right )}{4}\) | \(121\) |
trager | \(\text {Expression too large to display}\) | \(3627\) |
-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*(x^2*(x^2+1))^(1/4)/x*2^(3/4))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.60 (sec) , antiderivative size = 1149, normalized size of antiderivative = 10.17 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\text {Too large to display} \]
-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*I*8^(3 /4)*log(-(4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 + I*8^(3/4)*sqrt(x^4 + x^2)*x - 8^(1/4)*(3*I*x^3 + I*x) - 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 1/64*I*8^(3/4) *log(-(4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 - I*8^(3/4)*sqrt(x^4 + x^2)*x - 8^( 1/4)*(-3*I*x^3 - I*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*sqr t(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))*s qrt(-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) - (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))*sqr t(-sqrt(-I + 1)))/(x^5 + x)) - 1/16*sqrt(-sqrt(-I + 1))*log(-(4*sqrt(-I...
Not integrable
Time = 0.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.28 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int \frac {x^{4} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.47 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int { \frac {{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{4}}{x^{8} - 1} \,d x } \]
-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 = 0.32 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.93 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=-\frac {1}{4} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - 8 i \, \left (-\frac {1}{16777216} i + \frac {1}{16777216}\right )^{\frac {1}{4}} \log \left (\left (-4722366482869645213696 i + 4722366482869645213696\right )^{\frac {1}{4}} - 262144 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 8 i \, \left (-\frac {1}{16777216} i + \frac {1}{16777216}\right )^{\frac {1}{4}} \log \left (-\left (-4722366482869645213696 i + 4722366482869645213696\right )^{\frac {1}{4}} - 262144 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (\left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (i \, \left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-i \, \left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-\left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \left (-\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (i \, \left (-68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-i \, \left (-68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) \]
-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 = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx=\int \frac {x^4\,{\left (x^4+x^2\right )}^{1/4}}{x^8-1} \,d x \]