3.18.83 \(\int \frac {-1+x^4}{(1+x^4) \sqrt {1-x-x^2+x^3+x^4}} \, dx\) [1783]
Optimal. Leaf size=120 \[ \frac {1}{2} \text {RootSum}\left [3-8 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\& ,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}+\text {$\#$1}^3}\& \right ] \]
[Out]
Unintegrable
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Rubi [F]
time = 0.49, 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 {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-1 + x^4)/((1 + x^4)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]
[Out]
Defer[Int][1/Sqrt[1 - x - x^2 + x^3 + x^4], x] - ((-1)^(1/4)*Defer[Int][1/(((-1)^(1/4) - x)*Sqrt[1 - x - x^2 +
x^3 + x^4]), x])/2 + ((-1)^(3/4)*Defer[Int][1/((-(-1)^(3/4) - x)*Sqrt[1 - x - x^2 + x^3 + x^4]), x])/2 - ((-1
)^(1/4)*Defer[Int][1/(((-1)^(1/4) + x)*Sqrt[1 - x - x^2 + x^3 + x^4]), x])/2 + ((-1)^(3/4)*Defer[Int][1/((-(-1
)^(3/4) + x)*Sqrt[1 - x - x^2 + x^3 + x^4]), x])/2
Rubi steps
\begin {align*} \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {1-x-x^2+x^3+x^4}}-\frac {2}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ &=-\left (2 \int \left (\frac {i}{2 \left (i-x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}+\frac {i}{2 \left (i+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ &=-\left (i \int \frac {1}{\left (i-x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )-i \int \frac {1}{\left (i+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ &=-\left (i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {1-x-x^2+x^3+x^4}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx\right )-i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {1-x-x^2+x^3+x^4}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ &=-\left (\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )-\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 120, normalized size = 1.00 \begin {gather*} \frac {1}{2} \text {RootSum}\left [3-8 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-1 + x^4)/((1 + x^4)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]
[Out]
RootSum[3 - 8*#1 + 6*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1
^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-2 + 3*#1 + #1^3) & ]/2
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.98, size = 3623, normalized size = 30.19
| | |
| method |
result |
size |
| | |
| trager |
\(-\RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {12 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +8 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}+x^{3}-x^{2}-x +1}+3 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}+3 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x -3 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )+\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{12 x \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+x -1}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (\frac {12 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x -3 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}+48 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}+x^{3}-x^{2}-x +1}-\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +3 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )+2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}}{12 x \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-x^{2}+x +1}\right )}{6}\) |
\(398\) |
| default |
\(\text {Expression too large to display}\) |
\(3623\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(3623\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)
[Out]
2*(-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)+RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,inde
x=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,
index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)*(x-RootOf(_Z^4+_Z
^3-_Z^2-_Z+1,index=2))^2*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(
_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-Roo
tOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,inde
x=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,
index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3
-_Z^2-_Z+1,index=2))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/((x-RootOf(_Z^4
+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(x-R
ootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2)*EllipticF(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z
^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^
3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-Roo
tOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)+RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(
RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-
RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2))+1/9*sum(_alpha*(-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)+RootOf(_Z^4+
_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))^2/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf
(_Z^4+_Z^3-_Z^2-_Z+1,index=2))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(-1-_
alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)^3-2*_alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)^2+_alpha^3*RootOf
(_Z^4+_Z^3-_Z^2-_Z+1,index=2)+3*_alpha^3-_alpha^2*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)^3+2*RootOf(_Z^4+_Z^3-_Z^
2-_Z+1,index=2)*_alpha^2+_alpha^2+_alpha*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)^3+_alpha+RootOf(_Z^4+_Z^3-_Z^2-_Z
+1,index=2)^2*_alpha+2*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)+RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)^2)*((RootOf(_Z^
4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf
(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1
/2)*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i
ndex=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z
+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+
_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_
Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)/((x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,ind
ex=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2)*(3*EllipticF(((R
ootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)
)/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,inde
x=2)))^(1/2),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(-RootOf(_Z^4+_Z^3-_Z^
2-_Z+1,index=4)+RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^
2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2))+(RootOf(_Z^
4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(-1-_alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1
)^3-2*_alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^2+_alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)+3*_alpha^3-_
alpha^2*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^3+2*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)*_alpha^2+_alpha^2+_alpha*R
ootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^3+_alpha+RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^2*_alpha+2*RootOf(_Z^4+_Z^3-_Z
^2-_Z+1,index=1)+RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^2)*EllipticPi(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootO
f(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-R
ootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2),2/3*RootOf(_Z^4+_Z^3-_Z^2-_
Z+1,index=4)*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-1/3-2/3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^3-1/3*RootOf(_Z^4
+_Z^3-_Z^2-_Z+1,index=1)^2-2/3*_alpha+_alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)^3+5/3*_alpha^3*RootOf(_Z^4+
_Z^3-_Z^2-_Z+1,index=1)^2+4/3*_alpha^3*RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)+2/3*_alpha^2*RootOf(_Z^4+_Z^3-_Z^2-
_Z+1,index=1)^3+5/3*_alpha*RootOf(_Z^4+_Z^3-_Z^...
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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^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="maxima")
[Out]
integrate((x^4 - 1)/(sqrt(x^4 + x^3 - x^2 - x + 1)*(x^4 + 1)), x)
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.54, size = 4804, normalized size = 40.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="fricas")
[Out]
-1/24*3^(1/4)*sqrt(sqrt(3) + 3)*(sqrt(3) - 1)*log(3*(12*x^4 + 12*x^3 + 2*3^(1/4)*sqrt(x^4 + x^3 - x^2 - x + 1)
*(3*x^2 + sqrt(3)*(x^2 + 2*x - 1) - 3)*sqrt(sqrt(3) + 3) - 12*x^2 + 3*sqrt(3)*(3*x^4 + 4*x^3 - 4*x^2 - 4*x + 3
) - 12*x + 12)/(x^4 + 1)) + 1/24*3^(1/4)*sqrt(sqrt(3) + 3)*(sqrt(3) - 1)*log(3*(12*x^4 + 12*x^3 - 2*3^(1/4)*sq
rt(x^4 + x^3 - x^2 - x + 1)*(3*x^2 + sqrt(3)*(x^2 + 2*x - 1) - 3)*sqrt(sqrt(3) + 3) - 12*x^2 + 3*sqrt(3)*(3*x^
4 + 4*x^3 - 4*x^2 - 4*x + 3) - 12*x + 12)/(x^4 + 1)) - 1/6*3^(1/4)*sqrt(2)*sqrt(sqrt(3) + 3)*arctan(1/2*(27*sq
rt(3)*sqrt(2)*(7965*x^24 - 86940*x^23 - 452052*x^22 + 26692*x^21 + 2473150*x^20 + 1471532*x^19 - 6805092*x^18
- 5527220*x^17 + 12746227*x^16 + 11019368*x^15 - 18256392*x^14 - 15014808*x^13 + 20562084*x^12 + 15014808*x^11
- 18256392*x^10 - 11019368*x^9 + 12746227*x^8 + 5527220*x^7 - 6805092*x^6 - 1471532*x^5 + 2473150*x^4 - 26692
*x^3 - 452052*x^2 + 86940*x + 7965) + 6*sqrt(x^4 + x^3 - x^2 - x + 1)*(3*3^(3/4)*(sqrt(3)*sqrt(2)*(1251*x^22 +
53118*x^21 + 152273*x^20 - 226152*x^19 - 1135173*x^18 + 64070*x^17 + 3658401*x^16 + 1370976*x^15 - 7090226*x^
14 - 3772836*x^13 + 9542874*x^12 + 5019024*x^11 - 9542874*x^10 - 3772836*x^9 + 7090226*x^8 + 1370976*x^7 - 365
8401*x^6 + 64070*x^5 + 1135173*x^4 - 226152*x^3 - 152273*x^2 + 53118*x - 1251) - sqrt(2)*(14373*x^22 + 50208*x
^21 + 74455*x^20 + 37936*x^19 - 510931*x^18 - 1380336*x^17 + 818695*x^16 + 5137984*x^15 + 96914*x^14 - 9942384
*x^13 - 1619802*x^12 + 12190752*x^11 + 1619802*x^10 - 9942384*x^9 - 96914*x^8 + 5137984*x^7 - 818695*x^6 - 138
0336*x^5 + 510931*x^4 + 37936*x^3 - 74455*x^2 + 50208*x - 14373)) + 16*3^(1/4)*(sqrt(3)*sqrt(2)*(4028*x^22 + 2
1940*x^21 - 28445*x^20 - 217872*x^19 + 40111*x^18 + 938897*x^17 + 155918*x^16 - 2366280*x^15 - 698062*x^14 + 3
957631*x^13 + 1245100*x^12 - 4670064*x^11 - 1245100*x^10 + 3957631*x^9 + 698062*x^8 - 2366280*x^7 - 155918*x^6
+ 938897*x^5 - 40111*x^4 - 217872*x^3 + 28445*x^2 + 21940*x - 4028) - 3*sqrt(2)*(3472*x^22 + 8332*x^21 - 3281
2*x^20 - 80217*x^19 + 150191*x^18 + 366569*x^17 - 418571*x^16 - 1012572*x^15 + 777121*x^14 + 1821679*x^13 - 10
31713*x^12 - 2206854*x^11 + 1031713*x^10 + 1821679*x^9 - 777121*x^8 - 1012572*x^7 + 418571*x^6 + 366569*x^5 -
150191*x^4 - 80217*x^3 + 32812*x^2 + 8332*x - 3472)))*sqrt(sqrt(3) + 3) - sqrt(3)*(4*sqrt(x^4 + x^3 - x^2 - x
+ 1)*(432*sqrt(3)*sqrt(2)*(216*x^22 - 108*x^21 - 3330*x^20 - 781*x^19 + 19165*x^18 + 10195*x^17 - 59851*x^16 -
39556*x^15 + 118857*x^14 + 81429*x^13 - 163917*x^12 - 102318*x^11 + 163917*x^10 + 81429*x^9 - 118857*x^8 - 39
556*x^7 + 59851*x^6 + 10195*x^5 - 19165*x^4 - 781*x^3 + 3330*x^2 - 108*x - 216) + 32*sqrt(2)*(2360*x^22 - 8956
*x^21 - 45434*x^20 + 45495*x^19 + 282781*x^18 - 53897*x^17 - 907075*x^16 - 128232*x^15 + 1810289*x^14 + 486953
*x^13 - 2492597*x^12 - 681822*x^11 + 2492597*x^10 + 486953*x^9 - 1810289*x^8 - 128232*x^7 + 907075*x^6 - 53897
*x^5 - 282781*x^4 + 45495*x^3 + 45434*x^2 - 8956*x - 2360) + sqrt(3)*(sqrt(3)*sqrt(2)*(49986*x^22 - 21657*x^21
- 729134*x^20 - 244380*x^19 + 3857142*x^18 + 2321683*x^17 - 11301498*x^16 - 7948080*x^15 + 21680708*x^14 + 15
380790*x^13 - 29524188*x^12 - 18973992*x^11 + 29524188*x^10 + 15380790*x^9 - 21680708*x^8 - 7948080*x^7 + 1130
1498*x^6 + 2321683*x^5 - 3857142*x^4 - 244380*x^3 + 729134*x^2 - 21657*x - 49986) + 9*sqrt(2)*(7914*x^22 - 167
33*x^21 - 158670*x^20 + 22428*x^19 + 945222*x^18 + 334415*x^17 - 2948658*x^16 - 1612368*x^15 + 5841252*x^14 +
3501502*x^13 - 8053356*x^12 - 4458456*x^11 + 8053356*x^10 + 3501502*x^9 - 5841252*x^8 - 1612368*x^7 + 2948658*
x^6 + 334415*x^5 - 945222*x^4 + 22428*x^3 + 158670*x^2 - 16733*x - 7914))) - (3^(3/4)*(sqrt(3)*sqrt(2)*(74979*
x^24 + 33750*x^23 - 1015776*x^22 - 727438*x^21 + 5840466*x^20 + 4922886*x^19 - 19783936*x^18 - 17861550*x^17 +
44413005*x^16 + 40213276*x^15 - 70337952*x^14 - 59591436*x^13 + 81608124*x^12 + 59591436*x^11 - 70337952*x^10
- 40213276*x^9 + 44413005*x^8 + 17861550*x^7 - 19783936*x^6 - 4922886*x^5 + 5840466*x^4 + 727438*x^3 - 101577
6*x^2 - 33750*x + 74979) + 3*sqrt(2)*(35613*x^24 - 59994*x^23 - 860868*x^22 - 538030*x^21 + 4822062*x^20 + 551
6742*x^19 - 13789492*x^18 - 20062302*x^17 + 25455027*x^16 + 42469276*x^15 - 34669512*x^14 - 60232236*x^13 + 38
019396*x^12 + 60232236*x^11 - 34669512*x^10 - 42469276*x^9 + 25455027*x^8 + 20062302*x^7 - 13789492*x^6 - 5516
742*x^5 + 4822062*x^4 + 538030*x^3 - 860868*x^2 + 59994*x + 35613)) + 16*3^(1/4)*(sqrt(3)*sqrt(2)*(8748*x^24 +
6944*x^23 - 121117*x^22 - 160313*x^21 + 619191*x^20 + 1046203*x^19 - 1708067*x^18 - 3496102*x^17 + 3026016*x^
16 + 7267658*x^15 - 3929656*x^14 - 10279580*x^13 + 4209066*x^12 + 10279580*x^11 - 3929656*x^10 - 7267658*x^9 +
3026016*x^8 + 3496102*x^7 - 1708067*x^6 - 1046203*x^5 + 619191*x^4 + 160313*x^3 - 121117*x^2 - 6944*x + 8748)
+ 2*sqrt(2)*(3540*x^24 - 15692*x^23 - 97103*x^22 + 65354*x^21 + 679743*x^20 + 42464*x^19 - 2430064*x^18 - 835
238*x^17 + 5489208*x^16 + 2486254*x^15 - 8660693*x^14 - 3997426*x^13 + 10030362*x^12 + 3997426*x^11 - 8660693*
x^10 - 2486254*x^9 + 5489208*x^8 + 835238*x^7 -...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{4} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**4-1)/(x**4+1)/(x**4+x**3-x**2-x+1)**(1/2),x)
[Out]
Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**4 + 1)*sqrt(x**4 + x**3 - x**2 - x + 1)), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="giac")
[Out]
integrate((x^4 - 1)/(sqrt(x^4 + x^3 - x^2 - x + 1)*(x^4 + 1)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4-1}{\left (x^4+1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4 - 1)/((x^4 + 1)*(x^3 - x^2 - x + x^4 + 1)^(1/2)),x)
[Out]
int((x^4 - 1)/((x^4 + 1)*(x^3 - x^2 - x + x^4 + 1)^(1/2)), x)
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