3.28.61 \(\int \frac {(-1+x^2)^2 (x+x^3)}{\sqrt {1+x^4} (1-2 x^2+4 x^4-2 x^6+x^8)} \, dx\)

Optimal. Leaf size=260 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^7+12 \text {$\#$1}^6-4 \text {$\#$1}^5-10 \text {$\#$1}^4+4 \text {$\#$1}^3+12 \text {$\#$1}^2+4 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^6 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-2 \text {$\#$1}^5 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-7 \text {$\#$1}^4 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )+12 \text {$\#$1}^3 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )+7 \text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-2 \text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )-\log \left (-\text {$\#$1}+\sqrt {x^4+1}+x^2\right )}{2 \text {$\#$1}^7-7 \text {$\#$1}^6+18 \text {$\#$1}^5-5 \text {$\#$1}^4-10 \text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+1}\& \right ] \]

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Rubi [F]  time = 1.55, 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 {\left (-1+x^2\right )^2 \left (x+x^3\right )}{\sqrt {1+x^4} \left (1-2 x^2+4 x^4-2 x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^2)^2*(x + x^3))/(Sqrt[1 + x^4]*(1 - 2*x^2 + 4*x^4 - 2*x^6 + x^8)),x]

[Out]

Defer[Subst][Defer[Int][1/(Sqrt[1 + x^2]*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)), x], x, x^2]/2 - Defer[Subst][Defer[
Int][x/(Sqrt[1 + x^2]*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)), x], x, x^2]/2 - Defer[Subst][Defer[Int][x^2/(Sqrt[1 +
x^2]*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)), x], x, x^2]/2 + Defer[Subst][Defer[Int][x^3/(Sqrt[1 + x^2]*(1 - 2*x + 4
*x^2 - 2*x^3 + x^4)), x], x, x^2]/2

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right )^2 \left (x+x^3\right )}{\sqrt {1+x^4} \left (1-2 x^2+4 x^4-2 x^6+x^8\right )} \, dx &=\int \frac {x \left (-1+x^2\right )^2 \left (1+x^2\right )}{\sqrt {1+x^4} \left (1-2 x^2+4 x^4-2 x^6+x^8\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x)^2 (1+x)}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}-\frac {x}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}-\frac {x^2}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}+\frac {x^3}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \left (1-2 x+4 x^2-2 x^3+x^4\right )} \, dx,x,x^2\right )\\ \end {align*}

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Mathematica [C]  time = 7.48, size = 5697, normalized size = 21.91 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^2)^2*(x + x^3))/(Sqrt[1 + x^4]*(1 - 2*x^2 + 4*x^4 - 2*x^6 + x^8)),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.40, size = 274, normalized size = 1.05 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1-4 \text {$\#$1}+12 \text {$\#$1}^2-4 \text {$\#$1}^3-10 \text {$\#$1}^4+4 \text {$\#$1}^5+12 \text {$\#$1}^6+4 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-\log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right )+2 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}+7 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^2-12 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^3-7 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^5+\log \left (-x^2+\sqrt {1+x^4}-\text {$\#$1}\right ) \text {$\#$1}^6}{-1+6 \text {$\#$1}-3 \text {$\#$1}^2-10 \text {$\#$1}^3+5 \text {$\#$1}^4+18 \text {$\#$1}^5+7 \text {$\#$1}^6+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^2)^2*(x + x^3))/(Sqrt[1 + x^4]*(1 - 2*x^2 + 4*x^4 - 2*x^6 + x^8)),x]

[Out]

RootSum[1 - 4*#1 + 12*#1^2 - 4*#1^3 - 10*#1^4 + 4*#1^5 + 12*#1^6 + 4*#1^7 + #1^8 & , (-Log[-x^2 + Sqrt[1 + x^4
] - #1] + 2*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1 + 7*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^2 - 12*Log[-x^2 + Sqrt[1 +
x^4] - #1]*#1^3 - 7*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^4 + 2*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^5 + Log[-x^2 +
Sqrt[1 + x^4] - #1]*#1^6)/(-1 + 6*#1 - 3*#1^2 - 10*#1^3 + 5*#1^4 + 18*#1^5 + 7*#1^6 + 2*#1^7) & ]/4

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fricas [B]  time = 0.79, size = 455, normalized size = 1.75 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {x^{4} - x^{2} + \sqrt {2} {\left (x^{4} - x^{2} + 1\right )} - \sqrt {2 \, x^{8} - 2 \, x^{6} + 4 \, x^{4} - \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 2} {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )} + 1\right )} - \sqrt {x^{4} + 1} {\left (x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} - 1\right )} + 1}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x^{4} - x^{2} - \sqrt {2} {\left (x^{4} - x^{2} + 1\right )} - \sqrt {2 \, x^{8} - 2 \, x^{6} + 4 \, x^{4} + \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 2} {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 1\right )} - \sqrt {x^{4} + 1} {\left (x^{2} - \sqrt {2} {\left (x^{2} - 1\right )} - 1\right )} + 1}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (8 \, x^{8} - 8 \, x^{6} + 16 \, x^{4} + 4 \, \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - 4 \, {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 8\right ) + \frac {1}{8} \, \sqrt {2} \log \left (8 \, x^{8} - 8 \, x^{6} + 16 \, x^{4} - 4 \, \sqrt {2} {\left (x^{4} + x^{2} + 1\right )} - 4 \, {\left (2 \, x^{6} - 2 \, x^{4} + 3 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{4} + 1} + 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)^2*(x^3+x)/(x^4+1)^(1/2)/(x^8-2*x^6+4*x^4-2*x^2+1),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(-(x^4 - x^2 + sqrt(2)*(x^4 - x^2 + 1) - sqrt(2*x^8 - 2*x^6 + 4*x^4 - sqrt(2)*(x^4 + x^2 + 1
) - (2*x^6 - 2*x^4 + 3*x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(x^4 + 1) + 2)*(x^2 + sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1
)*(sqrt(2) + 1) + 1) - sqrt(x^4 + 1)*(x^2 + sqrt(2)*(x^2 - 1) - 1) + 1)/x^2) + 1/2*sqrt(2)*arctan((x^4 - x^2 -
sqrt(2)*(x^4 - x^2 + 1) - sqrt(2*x^8 - 2*x^6 + 4*x^4 + sqrt(2)*(x^4 + x^2 + 1) - (2*x^6 - 2*x^4 + 3*x^2 + sqr
t(2)*(x^2 + 1) + 1)*sqrt(x^4 + 1) + 2)*(x^2 - sqrt(2)*(x^2 + 1) - sqrt(x^4 + 1)*(sqrt(2) - 1) + 1) - sqrt(x^4
+ 1)*(x^2 - sqrt(2)*(x^2 - 1) - 1) + 1)/x^2) - 1/8*sqrt(2)*log(8*x^8 - 8*x^6 + 16*x^4 + 4*sqrt(2)*(x^4 + x^2 +
1) - 4*(2*x^6 - 2*x^4 + 3*x^2 + sqrt(2)*(x^2 + 1) + 1)*sqrt(x^4 + 1) + 8) + 1/8*sqrt(2)*log(8*x^8 - 8*x^6 + 1
6*x^4 - 4*sqrt(2)*(x^4 + x^2 + 1) - 4*(2*x^6 - 2*x^4 + 3*x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(x^4 + 1) + 8)

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giac [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^2-1)^2*(x^3+x)/(x^4+1)^(1/2)/(x^8-2*x^6+4*x^4-2*x^2+1),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 2.25, size = 67, normalized size = 0.26

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{3}-\textit {\_R}^{2}+\textit {\_R} +1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )-\textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{4}\)	\(67\)
elliptic	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{3}-\textit {\_R}^{2}+\textit {\_R} +1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )-\textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{4}\)	\(67\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x^{2} \sqrt {x^{4}+1}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-\sqrt {x^{4}+1}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+x^{4}-x^{2}+1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-x^{2} \sqrt {x^{4}+1}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+\sqrt {x^{4}+1}}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+x^{2}-1}\right )}{4}\)	\(275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)^2*(x^3+x)/(x^4+1)^(1/2)/(x^8-2*x^6+4*x^4-2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(_R*ln((-x^2+(x^4+1)^(1/2))^2+(_R^3-_R^2+_R+1)*(-x^2+(x^4+1)^(1/2))-_R^3+_R^2-_R),_R=RootOf(_Z^4+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + x\right )} {\left (x^{2} - 1\right )}^{2}}{{\left (x^{8} - 2 \, x^{6} + 4 \, x^{4} - 2 \, x^{2} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)^2*(x^3+x)/(x^4+1)^(1/2)/(x^8-2*x^6+4*x^4-2*x^2+1),x, algorithm="maxima")

[Out]

integrate((x^3 + x)*(x^2 - 1)^2/((x^8 - 2*x^6 + 4*x^4 - 2*x^2 + 1)*sqrt(x^4 + 1)), x)

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mupad [B]  time = 4.77, size = 475, normalized size = 1.83 \begin {gather*} \left (\sum _{k=1}^4\frac {{\left (\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )-1\right )}^2\,\left (\ln \left (-\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )+x^2\right )-\ln \left (\sqrt {\left ({\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1\right )\,\left (x^4+1\right )}+\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )\,x^2+1\right )\right )\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}{4\,\left (4\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )-3\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+2\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^3-1\right )\,\sqrt {{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1}}\right )+\left (\sum _{k=1}^4\frac {{\left (\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )-1\right )}^2\,\left (\ln \left (\sqrt {\left (x^4+1\right )\,\left ({\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1\right )}+x^2\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )+1\right )-\ln \left (x^2-\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )\right )\right )}{\sqrt {{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2+1}\,\left (-8\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^3+12\,{\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )}^2-16\,\mathrm {root}\left (z^4-2\,z^3+4\,z^2-2\,z+1,z,k\right )+4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 1)^2*(x + x^3))/((x^4 + 1)^(1/2)*(4*x^4 - 2*x^2 - 2*x^6 + x^8 + 1)),x)

[Out]

symsum(((root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k) - 1)^2*(log(x^2 - root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)
) - log(((root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)^2 + 1)*(x^4 + 1))^(1/2) + root(z^4 - 2*z^3 + 4*z^2 - 2*z +
1, z, k)*x^2 + 1))*root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k))/(4*(4*root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)
- 3*root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)^2 + 2*root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)^3 - 1)*(root(z^
4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)^2 + 1)^(1/2)), k, 1, 4) + symsum(((root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k
) - 1)^2*(log(((x^4 + 1)*(root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)^2 + 1))^(1/2) + x^2*root(z^4 - 2*z^3 + 4*z
^2 - 2*z + 1, z, k) + 1) - log(x^2 - root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k))))/((root(z^4 - 2*z^3 + 4*z^2 -
2*z + 1, z, k)^2 + 1)^(1/2)*(12*root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k)^2 - 8*root(z^4 - 2*z^3 + 4*z^2 - 2*
z + 1, z, k)^3 - 16*root(z^4 - 2*z^3 + 4*z^2 - 2*z + 1, z, k) + 4)), k, 1, 4)

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sympy [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**2-1)**2*(x**3+x)/(x**4+1)**(1/2)/(x**8-2*x**6+4*x**4-2*x**2+1),x)

[Out]

Timed out

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