∫ x4 _________________________________ (1+x4)2 4 √ ___

3.18.49 $\int \frac {x^4}{(1+x^4)^2 \sqrt [4]{x^2+x^4}} \, dx$ [1749]

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

[Out]

Unintegrable

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Rubi [F]
time = 0.08, 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 ) \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*}

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Mathematica [A]
time = 0.00, size = 138, normalized size = 1.17 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully veriﬁed.

[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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 103.04, size = 2554, normalized size = 21.64


method	result	size

risch	$\text {Expression too large to display}$	$2554$
trager	$\text {Expression too large to display}$	$2556$

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*(x^2-1)*x*(x^2+1)/(x^4+1)/(x^2*(x^2+1))^(1/4)+1/128*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*ln

((65536*x^3*RootOf(8192*_Z^8+128*_Z^4+1)^8*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)-65536*RootOf(_Z

^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*RootOf(8192*_Z^8+128*_Z^4+1)^8*x-512*(x^4+x^2)^(1/2)*RootOf(8192*_Z

^8+128*_Z^4+1)^4*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^3*x-1792*(x^4+x^2)^(1/4)*RootOf(8192*_Z^8

+128*_Z^4+1)^4*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^2*x^2-2304*RootOf(_Z^4+4096*RootOf(8192*_Z^

8+128*_Z^4+1)^4+64)*RootOf(8192*_Z^8+128*_Z^4+1)^4*x^3-6144*(x^4+x^2)^(3/4)*RootOf(8192*_Z^8+128*_Z^4+1)^4-281

6*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*RootOf(8192*_Z^8+128*_Z^4+1)^4*x-(x^4+x^2)^(1/2)*RootOf(

_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^3*x-16*(x^4+x^2)^(1/4)*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4

+1)^4+64)^2*x^2-36*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*x^3-112*(x^4+x^2)^(3/4)-24*RootOf(_Z^4+

4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*x)/(64*x^2*RootOf(8192*_Z^8+128*_Z^4+1)^4-64*RootOf(8192*_Z^8+128*_Z^4

+1)^4+x^2)/x)+8*RootOf(8192*_Z^8+128*_Z^4+1)^5*ln(-(16384*RootOf(8192*_Z^8+128*_Z^4+1)^9*x^3-16384*RootOf(8192

*_Z^8+128*_Z^4+1)^9*x+12288*RootOf(8192*_Z^8+128*_Z^4+1)^7*(x^4+x^2)^(1/2)*x+7168*(x^4+x^2)^(1/4)*RootOf(8192*

_Z^8+128*_Z^4+1)^6*x^2+1408*RootOf(8192*_Z^8+128*_Z^4+1)^5*x^3-384*(x^4+x^2)^(3/4)*RootOf(8192*_Z^8+128*_Z^4+1

)^4-128*RootOf(8192*_Z^8+128*_Z^4+1)^5*x-32*RootOf(8192*_Z^8+128*_Z^4+1)^3*(x^4+x^2)^(1/2)*x+48*(x^4+x^2)^(1/4

)*RootOf(8192*_Z^8+128*_Z^4+1)^2*x^2+24*RootOf(8192*_Z^8+128*_Z^4+1)*x^3+(x^4+x^2)^(3/4)+6*RootOf(8192*_Z^8+12

8*_Z^4+1)*x)/(64*x^2*RootOf(8192*_Z^8+128*_Z^4+1)^4-64*RootOf(8192*_Z^8+128*_Z^4+1)^4-1)/x)+1/16*RootOf(8192*_

Z^8+128*_Z^4+1)*ln(-(16384*RootOf(8192*_Z^8+128*_Z^4+1)^9*x^3-16384*RootOf(8192*_Z^8+128*_Z^4+1)^9*x+12288*Roo

tOf(8192*_Z^8+128*_Z^4+1)^7*(x^4+x^2)^(1/2)*x+7168*(x^4+x^2)^(1/4)*RootOf(8192*_Z^8+128*_Z^4+1)^6*x^2+1408*Roo

tOf(8192*_Z^8+128*_Z^4+1)^5*x^3-384*(x^4+x^2)^(3/4)*RootOf(8192*_Z^8+128*_Z^4+1)^4-128*RootOf(8192*_Z^8+128*_Z

^4+1)^5*x-32*RootOf(8192*_Z^8+128*_Z^4+1)^3*(x^4+x^2)^(1/2)*x+48*(x^4+x^2)^(1/4)*RootOf(8192*_Z^8+128*_Z^4+1)^

2*x^2+24*RootOf(8192*_Z^8+128*_Z^4+1)*x^3+(x^4+x^2)^(3/4)+6*RootOf(8192*_Z^8+128*_Z^4+1)*x)/(64*x^2*RootOf(819

2*_Z^8+128*_Z^4+1)^4-64*RootOf(8192*_Z^8+128*_Z^4+1)^4-1)/x)+ln((32768*x^3*RootOf(8192*_Z^8+128*_Z^4+1)^8*Root

Of(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)-32768*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*Root

Of(8192*_Z^8+128*_Z^4+1)^8*x-384*(x^4+x^2)^(1/2)*RootOf(8192*_Z^8+128*_Z^4+1)^4*RootOf(_Z^4+4096*RootOf(8192*_

Z^8+128*_Z^4+1)^4+64)^3*x+1792*(x^4+x^2)^(1/4)*RootOf(8192*_Z^8+128*_Z^4+1)^4*RootOf(_Z^4+4096*RootOf(8192*_Z^

8+128*_Z^4+1)^4+64)^2*x^2-1792*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*RootOf(8192*_Z^8+128*_Z^4+1

)^4*x^3-6144*(x^4+x^2)^(3/4)*RootOf(8192*_Z^8+128*_Z^4+1)^4-768*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^

4+64)*RootOf(8192*_Z^8+128*_Z^4+1)^4*x-7*(x^4+x^2)^(1/2)*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^3

*x+16*(x^4+x^2)^(1/4)*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^2*x^2+12*RootOf(_Z^4+4096*RootOf(819

2*_Z^8+128*_Z^4+1)^4+64)*x^3-112*(x^4+x^2)^(3/4)+8*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*x)/(64*

x^2*RootOf(8192*_Z^8+128*_Z^4+1)^4-64*RootOf(8192*_Z^8+128*_Z^4+1)^4+x^2)/x)*RootOf(8192*_Z^8+128*_Z^4+1)^4*Ro

otOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)+1/128*ln((32768*x^3*RootOf(8192*_Z^8+128*_Z^4+1)^8*RootOf(_Z

^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)-32768*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*RootOf(81

92*_Z^8+128*_Z^4+1)^8*x-384*(x^4+x^2)^(1/2)*RootOf(8192*_Z^8+128*_Z^4+1)^4*RootOf(_Z^4+4096*RootOf(8192*_Z^8+1

28*_Z^4+1)^4+64)^3*x+1792*(x^4+x^2)^(1/4)*RootOf(8192*_Z^8+128*_Z^4+1)^4*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128

*_Z^4+1)^4+64)^2*x^2-1792*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*RootOf(8192*_Z^8+128*_Z^4+1)^4*x

^3-6144*(x^4+x^2)^(3/4)*RootOf(8192*_Z^8+128*_Z^4+1)^4-768*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)

*RootOf(8192*_Z^8+128*_Z^4+1)^4*x-7*(x^4+x^2)^(1/2)*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^3*x+16

*(x^4+x^2)^(1/4)*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)^2*x^2+12*RootOf(_Z^4+4096*RootOf(8192*_Z^

8+128*_Z^4+1)^4+64)*x^3-112*(x^4+x^2)^(3/4)+8*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*_Z^4+1)^4+64)*x)/(64*x^2*R

ootOf(8192*_Z^8+128*_Z^4+1)^4-64*RootOf(8192*_Z^8+128*_Z^4+1)^4+x^2)/x)*RootOf(_Z^4+4096*RootOf(8192*_Z^8+128*

_Z^4+1)^4+64)-1/16*RootOf(8192*_Z^8+128*_Z^4+1)*ln(-(32768*RootOf(8192*_Z^8+128*_Z^4+1)^9*x^3-32768*RootOf(819

2*_Z^8+128*_Z^4+1)^9*x+16384*RootOf(8192*_Z^8+128*_Z^4+1)^7*(x^4+x^2)^(1/2)*x-7168*(x^4+x^2)^(1/4)*RootOf(8192

*_Z^8+128*_Z^4+1)^6*x^2+2176*RootOf(8192*_Z^8+128*_Z^4+1)^5*x^3-384*(x^4+x^2)^(3/4)*RootOf(8192*_Z^8+128*_Z^4+

1)^4+384*RootOf(8192*_Z^8+128*_Z^4+1)^5*x+224*RootOf(8192*_Z^8+128*_Z^4+1)^3*(x^4+x^2)^(1/2)*x-48*(x^4+x^2)^(1

/4)*RootOf(8192*_Z^8+128*_Z^4+1)^2*x^2+8*RootOf...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [F(-1)] Timed out
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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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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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.46, 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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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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3.18.49 \(\int \frac {x^4}{(1+x^4)^2 \sqrt [4]{x^2+x^4}} \, dx\) [1749]