Integrand size = 22, antiderivative size = 118 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {\left (-1+x^2\right ) \left (x^2+x^4\right )^{3/4}}{8 x \left (1+x^4\right )}+\frac {1}{64} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.17 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {16 x \left (-1+x^4\right )+\sqrt {x} \sqrt [4]{1+x^2} \left (1+x^4\right ) \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+4 \log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ]}{128 \left (1+x^4\right ) \sqrt [4]{x^2+x^4}} \]
(16*x*(-1 + x^4) + Sqrt[x]*(1 + x^2)^(1/4)*(1 + x^4)*RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 4*Log[(1 + x^2)^(1/4) - Sqrt[x]*#1] + Log[x]*#1^4 - 2*Log[(1 + x^2)^(1/4) - Sqrt[x]*#1]*#1^4)/(-#1 + #1^5) & ])/(128*(1 + x^4) *(x^2 + x^4)^(1/4))
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.
| \(\displaystyle \int \frac {x^4}{\left (x^4+1\right )^2 \sqrt [4]{x^4+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^{7/2}}{\sqrt [4]{x^2+1} \left (x^4+1\right )^2}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1593 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^4}{\sqrt [4]{x^2+1} \left (x^4+1\right )^2}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1888 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^4}{\sqrt [4]{x^2+1} \left (x^4+1\right )^2}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
3.18.49.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_ ), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[Int[x^(k*(m + 1 ) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)], x] ] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n) )^p, x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]
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 = 73.74 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}-\textit {\_R}}\right ) x^{5}+8 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {3}{4}} x^{2}-8 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {3}{4}}-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}-\textit {\_R}}\right ) x}{64 \left (x^{4}+1\right ) x}\) | \(151\) |
| trager | \(\text {Expression too large to display}\) | \(2552\) |
| risch | \(\text {Expression too large to display}\) | \(2559\) |
1/64*(-sum((_R^4-2)*ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(_R^5-_R),_R=RootOf( _Z^8-2*_Z^4+2))*x^5+8*(x^2*(x^2+1))^(3/4)*x^2-8*(x^2*(x^2+1))^(3/4)-sum((_ R^4-2)*ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(_R^5-_R),_R=RootOf(_Z^8-2*_Z^4+2 ))*x)/(x^4+1)/x
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.79 (sec) , antiderivative size = 1022, normalized size of antiderivative = 8.66 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \]
-1/128*((x^5 + x)*sqrt(sqrt(2)*sqrt(-I - 1))*log(-(2*sqrt(2)*sqrt(-I - 1)* ((I + 7)*x^4 + (7*I - 1)*x^2)*(x^4 + x^2)^(1/4) + 4*(x^4 + x^2)^(3/4)*(-(3 *I - 4)*x^2 + 4*I + 3) - (-(5*I - 15)*x^5 + (16*I + 12)*x^3 - 2*sqrt(2)*sq rt(-I - 1)*sqrt(x^4 + x^2)*(-(4*I + 3)*x^3 - (3*I - 4)*x) + (7*I - 1)*x)*s qrt(sqrt(2)*sqrt(-I - 1)))/(x^5 + x)) - (x^5 + x)*sqrt(sqrt(2)*sqrt(-I - 1 ))*log(-(2*sqrt(2)*sqrt(-I - 1)*((I + 7)*x^4 + (7*I - 1)*x^2)*(x^4 + x^2)^ (1/4) + 4*(x^4 + x^2)^(3/4)*(-(3*I - 4)*x^2 + 4*I + 3) - ((5*I - 15)*x^5 - (16*I + 12)*x^3 - 2*sqrt(2)*sqrt(-I - 1)*sqrt(x^4 + x^2)*((4*I + 3)*x^3 + (3*I - 4)*x) - (7*I - 1)*x)*sqrt(sqrt(2)*sqrt(-I - 1)))/(x^5 + x)) + (x^5 + x)*sqrt(sqrt(2)*sqrt(I - 1))*log(-(2*sqrt(2)*sqrt(I - 1)*(-(I - 7)*x^4 - (7*I + 1)*x^2)*(x^4 + x^2)^(1/4) + 4*(x^4 + x^2)^(3/4)*((3*I + 4)*x^2 - 4*I + 3) - ((5*I + 15)*x^5 - (16*I - 12)*x^3 - 2*sqrt(2)*sqrt(I - 1)*sqrt( x^4 + x^2)*((4*I - 3)*x^3 + (3*I + 4)*x) - (7*I + 1)*x)*sqrt(sqrt(2)*sqrt( I - 1)))/(x^5 + x)) - (x^5 + x)*sqrt(sqrt(2)*sqrt(I - 1))*log(-(2*sqrt(2)* sqrt(I - 1)*(-(I - 7)*x^4 - (7*I + 1)*x^2)*(x^4 + x^2)^(1/4) + 4*(x^4 + x^ 2)^(3/4)*((3*I + 4)*x^2 - 4*I + 3) - (-(5*I + 15)*x^5 + (16*I - 12)*x^3 - 2*sqrt(2)*sqrt(I - 1)*sqrt(x^4 + x^2)*(-(4*I - 3)*x^3 - (3*I + 4)*x) + (7* I + 1)*x)*sqrt(sqrt(2)*sqrt(I - 1)))/(x^5 + x)) + (x^5 + x)*sqrt(-sqrt(2)* sqrt(I - 1))*log(-(2*sqrt(2)*sqrt(I - 1)*(x^4 + x^2)^(1/4)*((I - 7)*x^4 + (7*I + 1)*x^2) + 4*(x^4 + x^2)^(3/4)*((3*I + 4)*x^2 - 4*I + 3) - ((5*I ...
Not integrable
Time = 1.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} + 1\right )^{2}}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{4}}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{2}} \,d x } \]
2/21*(4*x^5 + x^3 - 3*x)*x^(7/2)/((x^8 + 2*x^4 + 1)*(x^2 + 1)^(1/4)) - int egrate(16/21*(4*x^4 + x^2 - 3)*x^(7/2)/((x^12 + 3*x^8 + 3*x^4 + 1)*(x^2 + 1)^(1/4)), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.64 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {1}{2} i \, \left (-\frac {1}{524288} i - \frac {1}{524288}\right )^{\frac {1}{4}} \log \left (\left (248661618204893321077691124073410420050228075398673858720231988446579748506266687766528 i + 248661618204893321077691124073410420050228075398673858720231988446579748506266687766528\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 4722366482869645213696 i\right ) + \frac {1}{2} i \, \left (-\frac {1}{524288} i - \frac {1}{524288}\right )^{\frac {1}{4}} \log \left (-\left (248661618204893321077691124073410420050228075398673858720231988446579748506266687766528 i + 248661618204893321077691124073410420050228075398673858720231988446579748506266687766528\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 4722366482869645213696 i\right ) + \frac {1}{2} i \, \left (\frac {1}{524288} i - \frac {1}{524288}\right )^{\frac {1}{4}} \log \left (\left (-248661618204893321077691124073410420050228075398673858720231988446579748506266687766528 i + 248661618204893321077691124073410420050228075398673858720231988446579748506266687766528\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 4722366482869645213696 i\right ) - \frac {1}{2} i \, \left (\frac {1}{524288} i - \frac {1}{524288}\right )^{\frac {1}{4}} \log \left (-\left (-248661618204893321077691124073410420050228075398673858720231988446579748506266687766528 i + 248661618204893321077691124073410420050228075398673858720231988446579748506266687766528\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 4722366482869645213696 i\right ) - \frac {1}{8} \, \left (-\frac {1}{2048} i - \frac {1}{2048}\right )^{\frac {1}{4}} \log \left (i \, \left (187072209578355573530071658587684226515959365500928 i + 187072209578355573530071658587684226515959365500928\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 4398046511104 i\right ) + \frac {1}{8} \, \left (-\frac {1}{2048} i - \frac {1}{2048}\right )^{\frac {1}{4}} \log \left (-i \, \left (187072209578355573530071658587684226515959365500928 i + 187072209578355573530071658587684226515959365500928\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 4398046511104 i\right ) + \frac {1}{8} \, \left (\frac {1}{2048} i - \frac {1}{2048}\right )^{\frac {1}{4}} \log \left (i \, \left (-187072209578355573530071658587684226515959365500928 i + 187072209578355573530071658587684226515959365500928\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 4398046511104 i\right ) - \frac {1}{8} \, \left (\frac {1}{2048} i - \frac {1}{2048}\right )^{\frac {1}{4}} \log \left (-i \, \left (-187072209578355573530071658587684226515959365500928 i + 187072209578355573530071658587684226515959365500928\right )^{\frac {1}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 4398046511104 i\right ) - \frac {{\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{4}} - 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}}}{8 \, {\left ({\left (\frac {1}{x^{2}} + 1\right )}^{2} - \frac {2}{x^{2}}\right )}} \]
-1/2*I*(-1/524288*I - 1/524288)^(1/4)*log((2486616182048933210776911240734 10420050228075398673858720231988446579748506266687766528*I + 2486616182048 93321077691124073410420050228075398673858720231988446579748506266687766528 )^(1/4)*(1/x^2 + 1)^(1/4) + 4722366482869645213696*I) + 1/2*I*(-1/524288*I - 1/524288)^(1/4)*log(-(2486616182048933210776911240734104200502280753986 73858720231988446579748506266687766528*I + 2486616182048933210776911240734 10420050228075398673858720231988446579748506266687766528)^(1/4)*(1/x^2 + 1 )^(1/4) + 4722366482869645213696*I) + 1/2*I*(1/524288*I - 1/524288)^(1/4)* log((-24866161820489332107769112407341042005022807539867385872023198844657 9748506266687766528*I + 24866161820489332107769112407341042005022807539867 3858720231988446579748506266687766528)^(1/4)*(1/x^2 + 1)^(1/4) - 472236648 2869645213696*I) - 1/2*I*(1/524288*I - 1/524288)^(1/4)*log(-(-248661618204 89332107769112407341042005022807539867385872023198844657974850626668776652 8*I + 24866161820489332107769112407341042005022807539867385872023198844657 9748506266687766528)^(1/4)*(1/x^2 + 1)^(1/4) - 4722366482869645213696*I) - 1/8*(-1/2048*I - 1/2048)^(1/4)*log(I*(18707220957835557353007165858768422 6515959365500928*I + 187072209578355573530071658587684226515959365500928)^ (1/4)*(1/x^2 + 1)^(1/4) - 4398046511104*I) + 1/8*(-1/2048*I - 1/2048)^(1/4 )*log(-I*(187072209578355573530071658587684226515959365500928*I + 18707220 9578355573530071658587684226515959365500928)^(1/4)*(1/x^2 + 1)^(1/4) - ...
Not integrable
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,{\left (x^4+1\right )}^2} \,d x \]