Integrand size = 22, antiderivative size = 118 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {\left (-1+x^2\right ) \left (x^2+x^4\right )^{3/4}}{8 x \left (1+x^4\right )}+\frac {1}{64} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \] Output:
Unintegrable
Time = 0.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.17 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {16 x \left (-1+x^4\right )+\sqrt {x} \sqrt [4]{1+x^2} \left (1+x^4\right ) \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+4 \log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ]}{128 \left (1+x^4\right ) \sqrt [4]{x^2+x^4}} \] Input:
Integrate[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]
Output:
(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 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^4}{\sqrt [4]{x^2+1} \left (x^4+1\right )^2}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
Input:
Int[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]
Output:
$Aborted
Time = 84.50 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}-\textit {\_R}}\right ) x^{5}+8 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {3}{4}} x^{2}-8 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {3}{4}}-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}-\textit {\_R}}\right ) x}{64 x \left (x^{4}+1\right )}\) | \(151\) |
| trager | \(\text {Expression too large to display}\) | \(2553\) |
| risch | \(\text {Expression too large to display}\) | \(2560\) |
Input:
int(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x,method=_RETURNVERBOSE)
Output:
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/(x^4+1)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.57 (sec) , antiderivative size = 934, normalized size of antiderivative = 7.92 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="fricas")
Output:
-1/64*(sqrt(1/2)*(x^5 + x)*sqrt(-sqrt(1/2*I - 1/2))*log((sqrt(1/2)*((5*I + 15)*x^5 - (16*I - 12)*x^3 - 4*sqrt(1/2*I - 1/2)*sqrt(x^4 + x^2)*(-(4*I - 3)*x^3 - (3*I + 4)*x) - (7*I + 1)*x)*sqrt(-sqrt(1/2*I - 1/2)) - 2*sqrt(1/2 *I - 1/2)*(x^4 + x^2)^(1/4)*((I - 7)*x^4 + (7*I + 1)*x^2) - 2*(x^4 + x^2)^ (3/4)*((3*I + 4)*x^2 - 4*I + 3))/(x^5 + x)) - sqrt(1/2)*(x^5 + x)*sqrt(-sq rt(1/2*I - 1/2))*log((sqrt(1/2)*(-(5*I + 15)*x^5 + (16*I - 12)*x^3 - 4*sqr t(1/2*I - 1/2)*sqrt(x^4 + x^2)*((4*I - 3)*x^3 + (3*I + 4)*x) + (7*I + 1)*x )*sqrt(-sqrt(1/2*I - 1/2)) - 2*sqrt(1/2*I - 1/2)*(x^4 + x^2)^(1/4)*((I - 7 )*x^4 + (7*I + 1)*x^2) - 2*(x^4 + x^2)^(3/4)*((3*I + 4)*x^2 - 4*I + 3))/(x ^5 + x)) + sqrt(1/2)*(x^5 + x)*sqrt(-sqrt(-1/2*I - 1/2))*log((sqrt(1/2)*(- (5*I - 15)*x^5 + (16*I + 12)*x^3 - 4*sqrt(-1/2*I - 1/2)*sqrt(x^4 + x^2)*(( 4*I + 3)*x^3 + (3*I - 4)*x) + (7*I - 1)*x)*sqrt(-sqrt(-1/2*I - 1/2)) - 2*s qrt(-1/2*I - 1/2)*(x^4 + x^2)^(1/4)*(-(I + 7)*x^4 - (7*I - 1)*x^2) - 2*(x^ 4 + x^2)^(3/4)*(-(3*I - 4)*x^2 + 4*I + 3))/(x^5 + x)) - sqrt(1/2)*(x^5 + x )*sqrt(-sqrt(-1/2*I - 1/2))*log((sqrt(1/2)*((5*I - 15)*x^5 - (16*I + 12)*x ^3 - 4*sqrt(-1/2*I - 1/2)*sqrt(x^4 + x^2)*(-(4*I + 3)*x^3 - (3*I - 4)*x) - (7*I - 1)*x)*sqrt(-sqrt(-1/2*I - 1/2)) - 2*sqrt(-1/2*I - 1/2)*(x^4 + x^2) ^(1/4)*(-(I + 7)*x^4 - (7*I - 1)*x^2) - 2*(x^4 + x^2)^(3/4)*(-(3*I - 4)*x^ 2 + 4*I + 3))/(x^5 + x)) + sqrt(1/2)*(1/2*I - 1/2)^(1/4)*(x^5 + x)*log((sq rt(1/2)*(1/2*I - 1/2)^(1/4)*((5*I + 15)*x^5 - (16*I - 12)*x^3 - 4*sqrt(...
Not integrable
Time = 1.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} + 1\right )^{2}}\, dx \] Input:
integrate(x**4/(x**4+1)**2/(x**4+x**2)**(1/4),x)
Output:
Integral(x**4/((x**2*(x**2 + 1))**(1/4)*(x**4 + 1)**2), x)
Not integrable
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{4}}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{2}} \,d x } \] Input:
integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="maxima")
Output:
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 = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.64 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="giac")
Output:
-8*I*(-1/34359738368*I - 1/34359738368)^(1/4)*log((43934705024835902175884 16511412091659052438592091715462012456613878747637374499873358438170023330 91518546963929054774914375807231981865204004737810631363657728*I + 4393470 50248359021758841651141209165905243859209171546201245661387874763737449987 33584381700233309151854696392905477491437580723198186520400473781063136365 7728)^(1/4)*(1/x^2 + 1)^(1/4) + 5444517870735015415413993718908291383296*I ) + 8*I*(-1/34359738368*I - 1/34359738368)^(1/4)*log(-(4393470502483590217 58841651141209165905243859209171546201245661387874763737449987335843817002 333091518546963929054774914375807231981865204004737810631363657728*I + 439 34705024835902175884165114120916590524385920917154620124566138787476373744 99873358438170023330915185469639290547749143758072319818652040047378106313 63657728)^(1/4)*(1/x^2 + 1)^(1/4) + 54445178707350154154139937189082913832 96*I) - 1/2*(-1/524288*I - 1/524288)^(1/4)*log(I*(248661618204893321077691 124073410420050228075398673858720231988446579748506266687766528*I + 248661 61820489332107769112407341042005022807539867385872023198844657974850626668 7766528)^(1/4)*(1/x^2 + 1)^(1/4) - 4722366482869645213696*I) + 1/2*(-1/524 288*I - 1/524288)^(1/4)*log(-I*(248661618204893321077691124073410420050228 075398673858720231988446579748506266687766528*I + 248661618204893321077691 124073410420050228075398673858720231988446579748506266687766528)^(1/4)*(1/ x^2 + 1)^(1/4) - 4722366482869645213696*I) + 1/2*I*(1/524288*I - 1/5242...
Not integrable
Time = 7.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,{\left (x^4+1\right )}^2} \,d x \] Input:
int(x^4/((x^2 + x^4)^(1/4)*(x^4 + 1)^2),x)
Output:
int(x^4/((x^2 + x^4)^(1/4)*(x^4 + 1)^2), x)
Not integrable
Time = 0.49 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.86 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {-32 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {5}{4}} x^{4}-8 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {5}{4}} x^{2}-29 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {5}{4}}-117 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {1}{4}} x^{2}-117 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {1}{4}}+73 \sqrt {x^{2}+1}\, \left (\int \frac {\left (x^{2}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{10}+3 \sqrt {x}\, x^{8}+4 \sqrt {x}\, x^{6}+3 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{2}+\sqrt {x}}d x \right ) x^{6}+73 \sqrt {x^{2}+1}\, \left (\int \frac {\left (x^{2}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{10}+3 \sqrt {x}\, x^{8}+4 \sqrt {x}\, x^{6}+3 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{2}+\sqrt {x}}d x \right ) x^{4}+73 \sqrt {x^{2}+1}\, \left (\int \frac {\left (x^{2}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{10}+3 \sqrt {x}\, x^{8}+4 \sqrt {x}\, x^{6}+3 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{2}+\sqrt {x}}d x \right ) x^{2}+73 \sqrt {x^{2}+1}\, \left (\int \frac {\left (x^{2}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}+2 \sqrt {x}\, x^{10}+3 \sqrt {x}\, x^{8}+4 \sqrt {x}\, x^{6}+3 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{2}+\sqrt {x}}d x \right )+137 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x}{x^{12}+2 x^{10}+3 x^{8}+4 x^{6}+3 x^{4}+2 x^{2}+1}d x \right ) x^{6}+137 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x}{x^{12}+2 x^{10}+3 x^{8}+4 x^{6}+3 x^{4}+2 x^{2}+1}d x \right ) x^{4}+137 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x}{x^{12}+2 x^{10}+3 x^{8}+4 x^{6}+3 x^{4}+2 x^{2}+1}d x \right ) x^{2}+137 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x}{x^{12}+2 x^{10}+3 x^{8}+4 x^{6}+3 x^{4}+2 x^{2}+1}d x \right )}{351 \sqrt {x^{2}+1}\, \left (x^{6}+x^{4}+x^{2}+1\right )} \] Input:
int(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x)
Output:
( - 32*sqrt(x)*(x**2 + 1)**(5/4)*x**4 - 8*sqrt(x)*(x**2 + 1)**(5/4)*x**2 - 29*sqrt(x)*(x**2 + 1)**(5/4) - 117*sqrt(x)*(x**2 + 1)**(1/4)*x**2 - 117*s qrt(x)*(x**2 + 1)**(1/4) + 73*sqrt(x**2 + 1)*int((x**2 + 1)**(3/4)/(sqrt(x )*x**12 + 2*sqrt(x)*x**10 + 3*sqrt(x)*x**8 + 4*sqrt(x)*x**6 + 3*sqrt(x)*x* *4 + 2*sqrt(x)*x**2 + sqrt(x)),x)*x**6 + 73*sqrt(x**2 + 1)*int((x**2 + 1)* *(3/4)/(sqrt(x)*x**12 + 2*sqrt(x)*x**10 + 3*sqrt(x)*x**8 + 4*sqrt(x)*x**6 + 3*sqrt(x)*x**4 + 2*sqrt(x)*x**2 + sqrt(x)),x)*x**4 + 73*sqrt(x**2 + 1)*i nt((x**2 + 1)**(3/4)/(sqrt(x)*x**12 + 2*sqrt(x)*x**10 + 3*sqrt(x)*x**8 + 4 *sqrt(x)*x**6 + 3*sqrt(x)*x**4 + 2*sqrt(x)*x**2 + sqrt(x)),x)*x**2 + 73*sq rt(x**2 + 1)*int((x**2 + 1)**(3/4)/(sqrt(x)*x**12 + 2*sqrt(x)*x**10 + 3*sq rt(x)*x**8 + 4*sqrt(x)*x**6 + 3*sqrt(x)*x**4 + 2*sqrt(x)*x**2 + sqrt(x)),x ) + 137*sqrt(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)*x)/(x**12 + 2*x**10 + 3*x**8 + 4*x**6 + 3*x**4 + 2*x**2 + 1),x)*x**6 + 137*sqrt(x**2 + 1)*int( (sqrt(x)*(x**2 + 1)**(3/4)*x)/(x**12 + 2*x**10 + 3*x**8 + 4*x**6 + 3*x**4 + 2*x**2 + 1),x)*x**4 + 137*sqrt(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)* x)/(x**12 + 2*x**10 + 3*x**8 + 4*x**6 + 3*x**4 + 2*x**2 + 1),x)*x**2 + 137 *sqrt(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)*x)/(x**12 + 2*x**10 + 3*x** 8 + 4*x**6 + 3*x**4 + 2*x**2 + 1),x))/(351*sqrt(x**2 + 1)*(x**6 + x**4 + x **2 + 1))