3.18.49 \(\int \frac {x^4}{(1+x^4)^2 \sqrt [4]{x^2+x^4}} \, dx\)
Optimal. Leaf size=118 \[ \frac {1}{64} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-2 \log (x)}{\text {$\#$1}^5-\text {$\#$1}}\& \right ]+\frac {\left (x^4+x^2\right )^{3/4} \left (x^2-1\right )}{8 x \left (x^4+1\right )} \]
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Rubi [F] time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[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*} \int \frac {x^4}{\left (1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx &=\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 ) \operatorname {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*}
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Mathematica [C] time = 5.40, size = 148, normalized size = 1.25 \begin {gather*} \frac {x \left (\frac {4 \left (x^4-1\right )}{x^4+1}+\sqrt [4]{\frac {1}{x^2}+1} \left (-(1-i)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{1-i}}\right )-(1+i)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{1+i}}\right )+(1-i)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{1-i}}\right )+(1+i)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{1+i}}\right )\right )\right )}{32 \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]
[Out]
(x*((4*(-1 + x^4))/(1 + x^4) + (1 + x^(-2))^(1/4)*(-((1 - I)^(3/4)*ArcTan[(1 + x^(-2))^(1/4)/(1 - I)^(1/4)]) -
(1 + I)^(3/4)*ArcTan[(1 + x^(-2))^(1/4)/(1 + I)^(1/4)] + (1 - I)^(3/4)*ArcTanh[(1 + x^(-2))^(1/4)/(1 - I)^(1/
4)] + (1 + I)^(3/4)*ArcTanh[(1 + x^(-2))^(1/4)/(1 + I)^(1/4)])))/(32*(x^2 + x^4)^(1/4))
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IntegrateAlgebraic [A] time = 0.00, size = 118, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]
[Out]
((-1 + x^2)*(x^2 + x^4)^(3/4))/(8*x*(1 + x^4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(x^2 + x^4)^
(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-#1 + #1^5) & ]/64
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x, algorithm="fricas")
[Out]
Timed out
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giac [B] time = 0.36, size = 193, normalized size = 1.64 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [B] time = 103.17, size = 2552, normalized size = 21.63
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(2552\) |
| risch |
\(\text {Expression too large to display}\) |
\(2560\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(x^4+1)^2/(x^4+x^2)^(1/4),x,method=_RETURNVERBOSE)
[Out]
1/8*(-1+x)*(1+x)/x/(x^4+1)*(x^4+x^2)^(3/4)+1/2*RootOf(137438953472*_Z^8+524288*_Z^4+1)*ln((4398046511104*x^3*R
ootOf(137438953472*_Z^8+524288*_Z^4+1)^9-4398046511104*x*RootOf(137438953472*_Z^8+524288*_Z^4+1)^9+34359738368
*(x^4+x^2)^(1/2)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^7*x+1879048192*RootOf(137438953472*_Z^8+524288*_Z^4+1
)^6*(x^4+x^2)^(1/4)*x^2+71303168*RootOf(137438953472*_Z^8+524288*_Z^4+1)^5*x^3+1572864*(x^4+x^2)^(3/4)*RootOf(
137438953472*_Z^8+524288*_Z^4+1)^4+12582912*RootOf(137438953472*_Z^8+524288*_Z^4+1)^5*x+114688*(x^4+x^2)^(1/2)
*RootOf(137438953472*_Z^8+524288*_Z^4+1)^3*x+3072*RootOf(137438953472*_Z^8+524288*_Z^4+1)^2*(x^4+x^2)^(1/4)*x^
2+64*RootOf(137438953472*_Z^8+524288*_Z^4+1)*x^3-(x^4+x^2)^(3/4)+16*RootOf(137438953472*_Z^8+524288*_Z^4+1)*x)
/(262144*x^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-262144*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-1)/x)+
1/128*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*ln(-(-1099511627776*RootOf(_Z^4+16777
216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^8*x^3+2097152*(x^4+x
^2)^(1/2)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+
1)^4+64)^3*x+1099511627776*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953
472*_Z^8+524288*_Z^4+1)^8*x+7340032*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^2*RootO
f(137438953472*_Z^8+524288*_Z^4+1)^4*(x^4+x^2)^(1/4)*x^2+9437184*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8
+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*x^3+25165824*(x^4+x^2)^(3/4)*RootOf(1374389534
72*_Z^8+524288*_Z^4+1)^4+(x^4+x^2)^(1/2)*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^3*
x+11534336*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*
_Z^4+1)^4*x+16*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^2*(x^4+x^2)^(1/4)*x^2+36*Roo
tOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*x^3+112*(x^4+x^2)^(3/4)+24*RootOf(_Z^4+1677721
6*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*x)/(262144*x^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-26214
4*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+x^2)/x)-4096*ln((-549755813888*RootOf(_Z^4+16777216*RootOf(1374389
53472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^8*x^3+1572864*(x^4+x^2)^(1/2)*RootOf(1
37438953472*_Z^8+524288*_Z^4+1)^4*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^3*x+54975
5813888*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^
4+1)^8*x+7340032*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^2*RootOf(137438953472*_Z^8
+524288*_Z^4+1)^4*(x^4+x^2)^(1/4)*x^2+7340032*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+6
4)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*x^3-25165824*(x^4+x^2)^(3/4)*RootOf(137438953472*_Z^8+524288*_Z^4
+1)^4+7*(x^4+x^2)^(1/2)*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^3*x+3145728*RootOf(
_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*x+16*Roo
tOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^2*(x^4+x^2)^(1/4)*x^2-12*RootOf(_Z^4+16777216*
RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*x^3-112*(x^4+x^2)^(3/4)-8*RootOf(_Z^4+16777216*RootOf(1374389534
72*_Z^8+524288*_Z^4+1)^4+64)*x)/(262144*x^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-262144*RootOf(1374389534
72*_Z^8+524288*_Z^4+1)^4+x^2)/x)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*RootOf(_Z^4+16777216*RootOf(1374389
53472*_Z^8+524288*_Z^4+1)^4+64)-1/128*ln((-549755813888*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_
Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^8*x^3+1572864*(x^4+x^2)^(1/2)*RootOf(137438953472*_Z^8+52
4288*_Z^4+1)^4*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^3*x+549755813888*RootOf(_Z^4
+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^8*x+7340032*Ro
otOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*(
x^4+x^2)^(1/4)*x^2+7340032*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953
472*_Z^8+524288*_Z^4+1)^4*x^3-25165824*(x^4+x^2)^(3/4)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+7*(x^4+x^2)^(
1/2)*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^3*x+3145728*RootOf(_Z^4+16777216*RootO
f(137438953472*_Z^8+524288*_Z^4+1)^4+64)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4*x+16*RootOf(_Z^4+16777216*R
ootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)^2*(x^4+x^2)^(1/4)*x^2-12*RootOf(_Z^4+16777216*RootOf(137438953472
*_Z^8+524288*_Z^4+1)^4+64)*x^3-112*(x^4+x^2)^(3/4)-8*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4
+1)^4+64)*x)/(262144*x^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-262144*RootOf(137438953472*_Z^8+524288*_Z^4
+1)^4+x^2)/x)*RootOf(_Z^4+16777216*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4+64)+262144*RootOf(137438953472*_Z
^8+524288*_Z^4+1)^5*ln(-(2199023255552*x^3*RootOf(137438953472*_Z^8+524288*_Z^4+1)^9-2199023255552*x*RootOf(13
7438953472*_Z^8+524288*_Z^4+1)^9+25769803776*(x^4+x^2)^(1/2)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^7*x+18790
48192*RootOf(137438953472*_Z^8+524288*_Z^4+1)^6*(x^4+x^2)^(1/4)*x^2+46137344*RootOf(137438953472*_Z^8+524288*_
Z^4+1)^5*x^3-1572864*(x^4+x^2)^(3/4)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-4194304*RootOf(137438953472*_Z^
8+524288*_Z^4+1)^5*x-16384*(x^4+x^2)^(1/2)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^3*x+3072*RootOf(13743895347
2*_Z^8+524288*_Z^4+1)^2*(x^4+x^2)^(1/4)*x^2+192*RootOf(137438953472*_Z^8+524288*_Z^4+1)*x^3+(x^4+x^2)^(3/4)+48
*RootOf(137438953472*_Z^8+524288*_Z^4+1)*x)/(262144*x^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-262144*RootO
f(137438953472*_Z^8+524288*_Z^4+1)^4-1)/x)+1/2*RootOf(137438953472*_Z^8+524288*_Z^4+1)*ln(-(2199023255552*x^3*
RootOf(137438953472*_Z^8+524288*_Z^4+1)^9-2199023255552*x*RootOf(137438953472*_Z^8+524288*_Z^4+1)^9+2576980377
6*(x^4+x^2)^(1/2)*RootOf(137438953472*_Z^8+524288*_Z^4+1)^7*x+1879048192*RootOf(137438953472*_Z^8+524288*_Z^4+
1)^6*(x^4+x^2)^(1/4)*x^2+46137344*RootOf(137438953472*_Z^8+524288*_Z^4+1)^5*x^3-1572864*(x^4+x^2)^(3/4)*RootOf
(137438953472*_Z^8+524288*_Z^4+1)^4-4194304*RootOf(137438953472*_Z^8+524288*_Z^4+1)^5*x-16384*(x^4+x^2)^(1/2)*
RootOf(137438953472*_Z^8+524288*_Z^4+1)^3*x+3072*RootOf(137438953472*_Z^8+524288*_Z^4+1)^2*(x^4+x^2)^(1/4)*x^2
+192*RootOf(137438953472*_Z^8+524288*_Z^4+1)*x^3+(x^4+x^2)^(3/4)+48*RootOf(137438953472*_Z^8+524288*_Z^4+1)*x)
/(262144*x^2*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-262144*RootOf(137438953472*_Z^8+524288*_Z^4+1)^4-1)/x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left (4 \, x^{5} + x^{3} - 3 \, x\right )} x^{\frac {7}{2}}}{21 \, {\left (x^{8} + 2 \, x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} - \int \frac {16 \, {\left (4 \, x^{4} + x^{2} - 3\right )} x^{\frac {7}{2}}}{21 \, {\left (x^{12} + 3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,{\left (x^4+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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