\(\int \frac {x^4}{(1+x^4)^2 \sqrt [4]{x^2+x^4}} \, dx\) [1749]

Optimal result

Integrand size = 22, antiderivative size = 118 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {\left (-1+x^2\right ) \left (x^2+x^4\right )^{3/4}}{8 x \left (1+x^4\right )}+\frac {1}{64} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx \]

[In]

Int[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]

[Out]

(2*Sqrt[x]*(1 + x^2)^(1/4)*Defer[Subst][Defer[Int][x^8/((1 + x^4)^(1/4)*(1 + x^8)^2), x], x, Sqrt[x]])/(x^2 +
x^4)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^2} \left (1+x^4\right )^2} \, dx}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt [4]{1+x^4} \left (1+x^8\right )^2} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.17 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {16 x \left (-1+x^4\right )+\sqrt {x} \sqrt [4]{1+x^2} \left (1+x^4\right ) \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+4 \log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ]}{128 \left (1+x^4\right ) \sqrt [4]{x^2+x^4}} \]

[In]

Integrate[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]

[Out]

(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))

Maple [N/A] (verified)

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	\(\text {Expression too large to display}\)	\(2559\)

[In]

int(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

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 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \]

[In]

integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="fricas")

[Out]

-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)*
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)*sqr
t(-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)*sq
rt(-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 + 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(-sq
rt(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 + 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 - 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 - 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)) - 16*(x^4 + x^2)^(3/4)*(x^
2 - 1))/(x^5 + x)

Sympy [N/A]

Not integrable

Time = 1.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} + 1\right )^{2}}\, dx \]

[In]

integrate(x**4/(x**4+1)**2/(x**4+x**2)**(1/4),x)

[Out]

Integral(x**4/((x**2*(x**2 + 1))**(1/4)*(x**4 + 1)**2), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{4}}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{2}} \,d x } \]

[In]

integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="maxima")

[Out]

2/21*(4*x^5 + x^3 - 3*x)*x^(7/2)/((x^8 + 2*x^4 + 1)*(x^2 + 1)^(1/4)) - integrate(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)

Giac [C] (verification not implemented)

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 )}} \]

[In]

integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="giac")

[Out]

-1/2*I*(-1/524288*I - 1/524288)^(1/4)*log((2486616182048933210776911240734104200502280753986738587202319884465
79748506266687766528*I + 2486616182048933210776911240734104200502280753986738587202319884465797485062666877665
28)^(1/4)*(1/x^2 + 1)^(1/4) + 4722366482869645213696*I) + 1/2*I*(-1/524288*I - 1/524288)^(1/4)*log(-(248661618
204893321077691124073410420050228075398673858720231988446579748506266687766528*I + 248661618204893321077691124
073410420050228075398673858720231988446579748506266687766528)^(1/4)*(1/x^2 + 1)^(1/4) + 4722366482869645213696
*I) + 1/2*I*(1/524288*I - 1/524288)^(1/4)*log((-24866161820489332107769112407341042005022807539867385872023198
8446579748506266687766528*I + 24866161820489332107769112407341042005022807539867385872023198844657974850626668
7766528)^(1/4)*(1/x^2 + 1)^(1/4) - 4722366482869645213696*I) - 1/2*I*(1/524288*I - 1/524288)^(1/4)*log(-(-2486
61618204893321077691124073410420050228075398673858720231988446579748506266687766528*I + 2486616182048933210776
91124073410420050228075398673858720231988446579748506266687766528)^(1/4)*(1/x^2 + 1)^(1/4) - 47223664828696452
13696*I) - 1/8*(-1/2048*I - 1/2048)^(1/4)*log(I*(187072209578355573530071658587684226515959365500928*I + 18707
2209578355573530071658587684226515959365500928)^(1/4)*(1/x^2 + 1)^(1/4) - 4398046511104*I) + 1/8*(-1/2048*I -
1/2048)^(1/4)*log(-I*(187072209578355573530071658587684226515959365500928*I + 18707220957835557353007165858768
4226515959365500928)^(1/4)*(1/x^2 + 1)^(1/4) - 4398046511104*I) + 1/8*(1/2048*I - 1/2048)^(1/4)*log(I*(-187072
209578355573530071658587684226515959365500928*I + 187072209578355573530071658587684226515959365500928)^(1/4)*(
1/x^2 + 1)^(1/4) + 4398046511104*I) - 1/8*(1/2048*I - 1/2048)^(1/4)*log(-I*(-187072209578355573530071658587684
226515959365500928*I + 187072209578355573530071658587684226515959365500928)^(1/4)*(1/x^2 + 1)^(1/4) + 43980465
11104*I) - 1/8*((1/x^2 + 1)^(7/4) - 2*(1/x^2 + 1)^(3/4))/((1/x^2 + 1)^2 - 2/x^2)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,{\left (x^4+1\right )}^2} \,d x \]

[In]

int(x^4/((x^2 + x^4)^(1/4)*(x^4 + 1)^2),x)

[Out]

int(x^4/((x^2 + x^4)^(1/4)*(x^4 + 1)^2), x)