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

Optimal. Leaf size=120 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4+6 \text {$\#$1}^2-8 \text {$\#$1}+3\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1} x-x^2+\sqrt {x^4+x^3-x^2-x+1}+1\right )+\text {$\#$1}^2 (-\log (x))-\log \left (-\text {$\#$1} x-x^2+\sqrt {x^4+x^3-x^2-x+1}+1\right )+\log (x)}{\text {$\#$1}^3+3 \text {$\#$1}-2}\& \right ] \]

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Rubi [F]  time = 0.79, 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 [C]  time = 6.30, size = 13896, normalized size = 115.80 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [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]

IntegrateAlgebraic[(-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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fricas [B]  time = 2.92, size = 4804, normalized size = 40.03

result too large to display

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 - 2430064*x^6 - 42464*x^5 + 679743*x^4 - 65354*x^3 - 97103*x^2 +
15692*x + 3540)))*sqrt(sqrt(3) + 3))*sqrt((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)) + sqrt(2)*(1420273*x^24 + 4805532*x^23 - 6990684*x^22 - 56445924*x^21 - 43034298*x^20 + 21791
8452*x^19 + 346132212*x^18 - 428680524*x^17 - 1032893025*x^16 + 521844888*x^15 + 1831043496*x^14 - 492849576*x
^13 - 2191867756*x^12 + 492849576*x^11 + 1831043496*x^10 - 521844888*x^9 - 1032893025*x^8 + 428680524*x^7 + 34
6132212*x^6 - 217918452*x^5 - 43034298*x^4 + 56445924*x^3 - 6990684*x^2 - 4805532*x + 1420273) - 72*sqrt(3)*(s
qrt(3)*sqrt(2)*(834*x^24 + 28329*x^23 + 57653*x^22 - 228636*x^21 - 623670*x^20 + 704868*x^19 + 2721145*x^18 -
1036077*x^17 - 6845226*x^16 + 627714*x^15 + 11408354*x^14 + 112296*x^13 - 13436772*x^12 - 112296*x^11 + 114083
54*x^10 - 627714*x^9 - 6845226*x^8 + 1036077*x^7 + 2721145*x^6 - 704868*x^5 - 623670*x^4 + 228636*x^3 + 57653*
x^2 - 28329*x + 834) - sqrt(2)*(9582*x^24 + 23819*x^23 - 41279*x^22 - 134546*x^21 + 94356*x^20 + 385810*x^19 -
178747*x^18 - 779191*x^17 + 283986*x^16 + 1208534*x^15 - 367862*x^14 - 1491188*x^13 + 398424*x^12 + 1491188*x
^11 - 367862*x^10 - 1208534*x^9 + 283986*x^8 + 779191*x^7 - 178747*x^6 - 385810*x^5 + 94356*x^4 + 134546*x^3 -
41279*x^2 - 23819*x + 9582)))/(1246223*x^24 + 2293920*x^23 - 26548320*x^22 - 65287680*x^21 + 127861530*x^20 +
421996416*x^19 - 268833120*x^18 - 1372131744*x^17 + 254704353*x^16 + 2785350336*x^15 - 28287552*x^14 - 388748
2752*x^13 - 120055892*x^12 + 3887482752*x^11 - 28287552*x^10 - 2785350336*x^9 + 254704353*x^8 + 1372131744*x^7
- 268833120*x^6 - 421996416*x^5 + 127861530*x^4 + 65287680*x^3 - 26548320*x^2 - 2293920*x + 1246223)) - 1/6*3
^(1/4)*sqrt(2)*sqrt(sqrt(3) + 3)*arctan(-1/2*(27*sqrt(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 + 64
070*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 - 3658401*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 - 99423
84*x^9 - 96914*x^8 + 5137984*x^7 - 818695*x^6 - 1380336*x^5 + 510931*x^4 + 37936*x^3 - 74455*x^2 + 50208*x - 1
4373)) + 16*3^(1/4)*(sqrt(3)*sqrt(2)*(4028*x^22 + 21940*x^21 - 28445*x^20 - 217872*x^19 + 40111*x^18 + 938897*
x^17 + 155918*x^16 - 2366280*x^15 - 698062*x^14 + 3957631*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 - 32812*x^20 - 80217*x^19 + 150191*x^18 + 366569*x^17 - 418571*x^
16 - 1012572*x^15 + 777121*x^14 + 1821679*x^13 - 1031713*x^12 - 2206854*x^11 + 1031713*x^10 + 1821679*x^9 - 77
7121*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 - 1023
18*x^11 + 163917*x^10 + 81429*x^9 - 118857*x^8 - 39556*x^7 + 59851*x^6 + 10195*x^5 - 19165*x^4 - 781*x^3 + 333
0*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 + 15380790*x^13 - 29524188*x^12 - 18973992*x^11 + 29524188*x^10
+ 15380790*x^9 - 21680708*x^8 - 7948080*x^7 + 11301498*x^6 + 2321683*x^5 - 3857142*x^4 - 244380*x^3 + 729134*
x^2 - 21657*x - 49986) + 9*sqrt(2)*(7914*x^22 - 16733*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 - 16
733*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 - 1015776*x^2 - 33750*x + 74979) + 3*sqrt(2)*(35613*x^24 - 59994*x^
23 - 860868*x^22 - 538030*x^21 + 4822062*x^20 + 5516742*x^19 - 13789492*x^18 - 20062302*x^17 + 25455027*x^16 +
42469276*x^15 - 34669512*x^14 - 60232236*x^13 + 38019396*x^12 + 60232236*x^11 - 34669512*x^10 - 42469276*x^9
+ 25455027*x^8 + 20062302*x^7 - 13789492*x^6 - 5516742*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 + 1046
203*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 + 61
9191*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 - 835238*x^17 + 5489208*x^16 + 2486254*x^15 - 8660693*x^14 - 399
7426*x^13 + 10030362*x^12 + 3997426*x^11 - 8660693*x^10 - 2486254*x^9 + 5489208*x^8 + 835238*x^7 - 2430064*x^6
- 42464*x^5 + 679743*x^4 - 65354*x^3 - 97103*x^2 + 15692*x + 3540)))*sqrt(sqrt(3) + 3))*sqrt((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)) + sqrt(2)*(1420273*x^24 + 4805532*x^23 - 6
990684*x^22 - 56445924*x^21 - 43034298*x^20 + 217918452*x^19 + 346132212*x^18 - 428680524*x^17 - 1032893025*x^
16 + 521844888*x^15 + 1831043496*x^14 - 492849576*x^13 - 2191867756*x^12 + 492849576*x^11 + 1831043496*x^10 -
521844888*x^9 - 1032893025*x^8 + 428680524*x^7 + 346132212*x^6 - 217918452*x^5 - 43034298*x^4 + 56445924*x^3 -
6990684*x^2 - 4805532*x + 1420273) - 72*sqrt(3)*(sqrt(3)*sqrt(2)*(834*x^24 + 28329*x^23 + 57653*x^22 - 228636
*x^21 - 623670*x^20 + 704868*x^19 + 2721145*x^18 - 1036077*x^17 - 6845226*x^16 + 627714*x^15 + 11408354*x^14 +
112296*x^13 - 13436772*x^12 - 112296*x^11 + 11408354*x^10 - 627714*x^9 - 6845226*x^8 + 1036077*x^7 + 2721145*
x^6 - 704868*x^5 - 623670*x^4 + 228636*x^3 + 57653*x^2 - 28329*x + 834) - sqrt(2)*(9582*x^24 + 23819*x^23 - 41
279*x^22 - 134546*x^21 + 94356*x^20 + 385810*x^19 - 178747*x^18 - 779191*x^17 + 283986*x^16 + 1208534*x^15 - 3
67862*x^14 - 1491188*x^13 + 398424*x^12 + 1491188*x^11 - 367862*x^10 - 1208534*x^9 + 283986*x^8 + 779191*x^7 -
178747*x^6 - 385810*x^5 + 94356*x^4 + 134546*x^3 - 41279*x^2 - 23819*x + 9582)))/(1246223*x^24 + 2293920*x^23
- 26548320*x^22 - 65287680*x^21 + 127861530*x^20 + 421996416*x^19 - 268833120*x^18 - 1372131744*x^17 + 254704
353*x^16 + 2785350336*x^15 - 28287552*x^14 - 3887482752*x^13 - 120055892*x^12 + 3887482752*x^11 - 28287552*x^1
0 - 2785350336*x^9 + 254704353*x^8 + 1372131744*x^7 - 268833120*x^6 - 421996416*x^5 + 127861530*x^4 + 65287680
*x^3 - 26548320*x^2 - 2293920*x + 1246223))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} + 1\right )}}\,{d x} \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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maple [B]  time = 1.95, size = 397, normalized size = 3.31

method	result	size
trager	\(\RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-12 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +8 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}+x^{3}-x^{2}-x +1}-3 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}-3 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x +\sqrt {x^{4}+x^{3}-x^{2}-x +1}+3 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )}{12 x \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+x -1}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (\frac {12 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +48 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}+x^{3}-x^{2}-x +1}-3 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+3 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )}{12 x \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-x^{2}+x +1}\right )}{6}\)	\(397\)
default	\(\text {Expression too large to display}\)	\(3623\)
elliptic	\(\text {Expression too large to display}\)	\(3623\)

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]

RootOf(48*_Z^4+8*_Z^2+1)*ln(-(-12*RootOf(48*_Z^4+8*_Z^2+1)^3*x+8*RootOf(48*_Z^4+8*_Z^2+1)^2*(x^4+x^3-x^2-x+1)^
(1/2)-3*RootOf(48*_Z^4+8*_Z^2+1)*x^2-3*RootOf(48*_Z^4+8*_Z^2+1)*x+(x^4+x^3-x^2-x+1)^(1/2)+3*RootOf(48*_Z^4+8*_
Z^2+1))/(12*x*RootOf(48*_Z^4+8*_Z^2+1)^2+x^2+x-1))-1/6*RootOf(_Z^2+36*RootOf(48*_Z^4+8*_Z^2+1)^2+6)*ln((12*Roo
tOf(48*_Z^4+8*_Z^2+1)^2*RootOf(_Z^2+36*RootOf(48*_Z^4+8*_Z^2+1)^2+6)*x+48*RootOf(48*_Z^4+8*_Z^2+1)^2*(x^4+x^3-
x^2-x+1)^(1/2)-3*RootOf(_Z^2+36*RootOf(48*_Z^4+8*_Z^2+1)^2+6)*x^2-RootOf(_Z^2+36*RootOf(48*_Z^4+8*_Z^2+1)^2+6)
*x+2*(x^4+x^3-x^2-x+1)^(1/2)+3*RootOf(_Z^2+36*RootOf(48*_Z^4+8*_Z^2+1)^2+6))/(12*x*RootOf(48*_Z^4+8*_Z^2+1)^2-
x^2+x+1))

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