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

Optimal. Leaf size=99 \[ \sqrt {\frac {1}{5}-\frac {3 i}{5}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {2 x^2-x-2}{x^2+x-1}}}{\sqrt {-1-3 i}}\right )+\sqrt {\frac {1}{5}+\frac {3 i}{5}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {2 x^2-x-2}{x^2+x-1}}}{\sqrt {-1+3 i}}\right ) \]

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Rubi [F]  time = 4.18, 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^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/2*((1 + I*Sqrt[3])*Sqrt[-2 - x + 2*x^2]*Defer[Int][Sqrt[-1 + x + x^2]/((Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x)*Sq
rt[-2 - x + 2*x^2]), x])/(Sqrt[1 - I*Sqrt[3]]*Sqrt[(2 + x - 2*x^2)/(1 - x - x^2)]*Sqrt[-1 + x + x^2]) - ((1 -
I*Sqrt[3])*Sqrt[-2 - x + 2*x^2]*Defer[Int][Sqrt[-1 + x + x^2]/((Sqrt[1 + I*Sqrt[3]] - Sqrt[2]*x)*Sqrt[-2 - x +
 2*x^2]), x])/(2*Sqrt[1 + I*Sqrt[3]]*Sqrt[(2 + x - 2*x^2)/(1 - x - x^2)]*Sqrt[-1 + x + x^2]) - ((1 + I*Sqrt[3]
)*Sqrt[-2 - x + 2*x^2]*Defer[Int][Sqrt[-1 + x + x^2]/((Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x)*Sqrt[-2 - x + 2*x^2]),
 x])/(2*Sqrt[1 - I*Sqrt[3]]*Sqrt[(2 + x - 2*x^2)/(1 - x - x^2)]*Sqrt[-1 + x + x^2]) - ((1 - I*Sqrt[3])*Sqrt[-2
 - x + 2*x^2]*Defer[Int][Sqrt[-1 + x + x^2]/((Sqrt[1 + I*Sqrt[3]] + Sqrt[2]*x)*Sqrt[-2 - x + 2*x^2]), x])/(2*S
qrt[1 + I*Sqrt[3]]*Sqrt[(2 + x - 2*x^2)/(1 - x - x^2)]*Sqrt[-1 + x + x^2])

Rubi steps

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

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Mathematica [C]  time = 6.72, size = 11770, normalized size = 118.89 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.49, size = 97, normalized size = 0.98 \begin {gather*} \sqrt {\frac {1}{5}+\frac {3 i}{5}} \tan ^{-1}\left (\sqrt {-\frac {1}{5}-\frac {3 i}{5}} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}\right )+\sqrt {\frac {1}{5}-\frac {3 i}{5}} \tan ^{-1}\left (\sqrt {-\frac {1}{5}+\frac {3 i}{5}} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1/5 + (3*I)/5]*ArcTan[Sqrt[-1/5 - (3*I)/5]*Sqrt[(-2 - x + 2*x^2)/(-1 + x + x^2)]] + Sqrt[1/5 - (3*I)/5]*A
rcTan[Sqrt[-1/5 + (3*I)/5]*Sqrt[(-2 - x + 2*x^2)/(-1 + x + x^2)]]

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fricas [B]  time = 6.55, size = 5095, normalized size = 51.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/100*10^(3/4)*sqrt(5)*sqrt(2*sqrt(10) + 20)*arctan(1/900*(113784843418327559700*x^24 - 163924417908538346160
0*x^23 + 2545884655734590980200*x^22 + 50821242952569317006400*x^21 - 158890131031597690275900*x^20 - 46505023
7434908643152000*x^19 + 1504931893222349028684600*x^18 + 2129607736390894049244000*x^17 - 62236607711339273673
15000*x^16 - 5602507922355181669879200*x^15 + 13883484083974894847147400*x^14 + 8971054092806414576313600*x^13
 - 18019091872612617624668700*x^12 - 8971054092806414576313600*x^11 + 13883484083974894847147400*x^10 + 560250
7922355181669879200*x^9 - 6223660771133927367315000*x^8 - 2129607736390894049244000*x^7 + 15049318932223490286
84600*x^6 + 465050237434908643152000*x^5 - 158890131031597690275900*x^4 - 50821242952569317006400*x^3 + 254588
4655734590980200*x^2 + 17*sqrt(2)*((10^(3/4)*(sqrt(10)*sqrt(5)*(3353156233262299*x^24 - 49929701526135692*x^23
 + 105488376266631398*x^22 + 1245603618916311800*x^21 - 3729015017341775649*x^20 - 12574465648522249504*x^19 +
 28773197469313801650*x^18 + 61934497275400333884*x^17 - 104756202796838736914*x^16 - 169570606189065646500*x^
15 + 217084068264548775966*x^14 + 276610136446380234320*x^13 - 274989879451094143625*x^12 - 276610136446380234
320*x^11 + 217084068264548775966*x^10 + 169570606189065646500*x^9 - 104756202796838736914*x^8 - 61934497275400
333884*x^7 + 28773197469313801650*x^6 + 12574465648522249504*x^5 - 3729015017341775649*x^4 - 12456036189163118
00*x^3 + 105488376266631398*x^2 + 49929701526135692*x + 3353156233262299) + 10*sqrt(5)*(1603095924058903*x^24
- 28424949163733294*x^23 + 129882833582498276*x^22 + 237455573469526820*x^21 - 2351987676228007923*x^20 - 4789
83875485473448*x^19 + 15731365763207610420*x^18 - 1230955405989439242*x^17 - 54757871112862546478*x^16 + 75442
91365549818150*x^15 + 111831054251172443652*x^14 - 14992902143911637200*x^13 - 141157732738263834575*x^12 + 14
992902143911637200*x^11 + 111831054251172443652*x^10 - 7544291365549818150*x^9 - 54757871112862546478*x^8 + 12
30955405989439242*x^7 + 15731365763207610420*x^6 + 478983875485473448*x^5 - 2351987676228007923*x^4 - 23745557
3469526820*x^3 + 129882833582498276*x^2 + 28424949163733294*x + 1603095924058903)) + 160*10^(1/4)*(2*sqrt(10)*
sqrt(5)*(25826047493168*x^24 - 423236377284370*x^23 + 1395910997963665*x^22 + 7977814049523724*x^21 - 39442227
900912726*x^20 - 55889144962836926*x^19 + 308168334219135075*x^18 + 191213417542285986*x^17 - 1166656659447404
872*x^16 - 371314532640925644*x^15 + 2495683096606018509*x^14 + 476597108839388734*x^13 - 3197719822327310998*
x^12 - 476597108839388734*x^11 + 2495683096606018509*x^10 + 371314532640925644*x^9 - 1166656659447404872*x^8 -
 191213417542285986*x^7 + 308168334219135075*x^6 + 55889144962836926*x^5 - 39442227900912726*x^4 - 79778140495
23724*x^3 + 1395910997963665*x^2 + 423236377284370*x + 25826047493168) + sqrt(5)*(167291197847878*x^24 - 28217
79357503405*x^23 + 10956936121559405*x^22 + 35249787921032759*x^21 - 204017758359407706*x^20 - 312436240830040
741*x^19 + 1399334424550385115*x^18 + 1735800349150666551*x^17 - 4987682548518497012*x^16 - 536838175952258778
9*x^15 + 10387543967266594209*x^14 + 9352242816540595199*x^13 - 13220291544952190258*x^12 - 935224281654059519
9*x^11 + 10387543967266594209*x^10 + 5368381759522587789*x^9 - 4987682548518497012*x^8 - 1735800349150666551*x
^7 + 1399334424550385115*x^6 + 312436240830040741*x^5 - 204017758359407706*x^4 - 35249787921032759*x^3 + 10956
936121559405*x^2 + 2821779357503405*x + 167291197847878)))*sqrt(2*sqrt(10) + 20) - 40*(7575923063334080*x^24 -
 120846800036516320*x^23 + 365161535455038400*x^22 + 2264233628233167520*x^21 - 8547268899540342720*x^20 - 247
22670274627958240*x^19 + 64588595464717514880*x^18 + 139472668675247371680*x^17 - 248284848145133462080*x^16 -
 427713008722866409440*x^15 + 544910109896139954240*x^14 + 743321346231612491680*x^13 - 707219317504256947840*
x^12 - 743321346231612491680*x^11 + 544910109896139954240*x^10 + 427713008722866409440*x^9 - 24828484814513346
2080*x^8 - 139472668675247371680*x^7 + 64588595464717514880*x^6 + 24722670274627958240*x^5 - 85472688995403427
20*x^4 - 2264233628233167520*x^3 + 365161535455038400*x^2 + sqrt(10)*(3941811839020250*x^24 - 6434846577568855
0*x^23 + 225885647210950450*x^22 + 941020417723504900*x^21 - 4570387865727921300*x^20 - 7999664382660162200*x^
19 + 29582666125955145150*x^18 + 37508204562313810350*x^17 - 97777371150634079200*x^16 - 100481781887576998050
*x^15 + 191989493413612300650*x^14 + 162041691927783863200*x^13 - 238833086260082653000*x^12 - 162041691927783
863200*x^11 + 191989493413612300650*x^10 + 100481781887576998050*x^9 - 97777371150634079200*x^8 - 375082045623
13810350*x^7 + 29582666125955145150*x^6 + 7999664382660162200*x^5 - 4570387865727921300*x^4 - 9410204177235049
00*x^3 + 225885647210950450*x^2 + sqrt(10)*(1160866886007179*x^24 - 17845297455339433*x^23 + 44598386983244311
*x^22 + 407427605373345862*x^21 - 1374776119098044166*x^20 - 4157698868742924356*x^19 + 10927521516291262449*x
^18 + 21652775358983721237*x^17 - 41131357142108902096*x^16 - 62264224681051657251*x^15 + 87175766672417705979
*x^14 + 104283289775600539600*x^13 - 111301173832927037212*x^12 - 104283289775600539600*x^11 + 871757666724177
05979*x^10 + 62264224681051657251*x^9 - 41131357142108902096*x^8 - 21652775358983721237*x^7 + 1092752151629126
2449*x^6 + 4157698868742924356*x^5 - 1374776119098044166*x^4 - 407427605373345862*x^3 + 44598386983244311*x^2
+ 17845297455339433*x + 1160866886007179) + 64348465775688550*x + 3941811839020250) + 462400*sqrt(10)*(5159465
558*x^24 - 81725010367*x^23 + 233461919938*x^22 + 1698386911873*x^21 - 6557645525970*x^20 - 15828018471263*x^1
9 + 48718561372260*x^18 + 76722347870157*x^17 - 172301271560638*x^16 - 208531808984739*x^15 + 348922649378334*
x^14 + 338140378140253*x^13 - 438000414906964*x^12 - 338140378140253*x^11 + 348922649378334*x^10 + 20853180898
4739*x^9 - 172301271560638*x^8 - 76722347870157*x^7 + 48718561372260*x^6 + 15828018471263*x^5 - 6557645525970*
x^4 - 1698386911873*x^3 + 233461919938*x^2 + 81725010367*x + 5159465558) + 120846800036516320*x + 757592306333
4080)*sqrt((2*x^2 - x - 2)/(x^2 + x - 1)))*sqrt((400*x^4 + 200*x^3 + 2*10^(1/4)*(sqrt(10)*sqrt(5)*(2*x^4 + x^3
 - 5*x^2 - x + 2) + 5*sqrt(5)*(x^4 + 2*x^3 - x^2 - 2*x + 1))*sqrt(2*sqrt(10) + 20)*sqrt((2*x^2 - x - 2)/(x^2 +
 x - 1)) - 1000*x^2 + 5*sqrt(10)*(17*x^4 + 4*x^3 - 29*x^2 - 4*x + 17) - 200*x + 400)/(x^4 - x^2 + 1)) + 86700*
(10^(3/4)*(sqrt(10)*sqrt(5)*(2442716712885*x^24 - 13728698401842*x^23 - 293186299957362*x^22 + 231013736762196
4*x^21 + 1451517102418989*x^20 - 33418412281395816*x^19 - 7178198464024966*x^18 + 183000238657163274*x^17 + 32
979923853964410*x^16 - 518819784198533318*x^15 - 84251682815179386*x^14 + 855052654520890752*x^13 + 1146378258
72128285*x^12 - 855052654520890752*x^11 - 84251682815179386*x^10 + 518819784198533318*x^9 + 32979923853964410*
x^8 - 183000238657163274*x^7 - 7178198464024966*x^6 + 33418412281395816*x^5 + 1451517102418989*x^4 - 231013736
7621964*x^3 - 293186299957362*x^2 + 13728698401842*x + 2442716712885) + 2*sqrt(5)*(14773174320516*x^24 - 26837
5520761317*x^23 + 1309307163746259*x^22 + 1740271771554118*x^21 - 22961098057094619*x^20 + 1995670027514196*x^
19 + 159995361247050941*x^18 - 36026323622261007*x^17 - 580081963181949234*x^16 + 111975250170214681*x^15 + 12
16656271908709911*x^14 - 182294736937952040*x^13 - 1549896649785392773*x^12 + 182294736937952040*x^11 + 121665
6271908709911*x^10 - 111975250170214681*x^9 - 580081963181949234*x^8 + 36026323622261007*x^7 + 159995361247050
941*x^6 - 1995670027514196*x^5 - 22961098057094619*x^4 - 1740271771554118*x^3 + 1309307163746259*x^2 + 2683755
20761317*x + 14773174320516)) + 32*10^(1/4)*(sqrt(10)*sqrt(5)*(353203013202*x^24 - 5252268148153*x^23 + 959989
9316880*x^22 + 157318487399903*x^21 - 528463660887158*x^20 - 1352806046560081*x^19 + 4754204159940742*x^18 + 5
302733573770307*x^17 - 19750563646089182*x^16 - 12045907326764941*x^15 + 44826810647943796*x^14 + 177720098479
98347*x^13 - 58660810041591536*x^12 - 17772009847998347*x^11 + 44826810647943796*x^10 + 12045907326764941*x^9
- 19750563646089182*x^8 - 5302733573770307*x^7 + 4754204159940742*x^6 + 1352806046560081*x^5 - 528463660887158
*x^4 - 157318487399903*x^3 + 9599899316880*x^2 + 5252268148153*x + 353203013202) + 5*sqrt(5)*(239278797030*x^2
4 - 3894104051149*x^23 + 14241545124129*x^22 + 43944733024733*x^21 - 190384021067588*x^20 - 581008343105785*x^
19 + 922766399143729*x^18 + 4154684453315363*x^17 - 1978225802691326*x^16 - 14164309049142511*x^15 + 216593389
5533713*x^14 + 25503495102958511*x^13 - 1877306138593118*x^12 - 25503495102958511*x^11 + 2165933895533713*x^10
 + 14164309049142511*x^9 - 1978225802691326*x^8 - 4154684453315363*x^7 + 922766399143729*x^6 + 581008343105785
*x^5 - 190384021067588*x^4 - 43944733024733*x^3 + 14241545124129*x^2 + 3894104051149*x + 239278797030)))*sqrt(
2*sqrt(10) + 20)*sqrt((2*x^2 - x - 2)/(x^2 + x - 1)) + 1300500*sqrt(10)*(33568388558203*x^24 - 575319820414644
*x^23 + 2600060853588106*x^22 + 3231189194065640*x^21 - 35687236548135733*x^20 - 7645624101059968*x^19 + 18267
8402540331150*x^18 + 9974894492537668*x^17 - 512039294651286218*x^16 - 7794844652266140*x^15 + 909771845956721
522*x^14 + 3856229278651280*x^13 - 1094808769706185685*x^12 - 3856229278651280*x^11 + 909771845956721522*x^10
+ 7794844652266140*x^9 - 512039294651286218*x^8 - 9974894492537668*x^7 + 182678402540331150*x^6 + 764562410105
9968*x^5 - 35687236548135733*x^4 - 3231189194065640*x^3 + 2600060853588106*x^2 + 575319820414644*x + 335683885
58203) - 10404000*sqrt(10)*(3913944654568*x^24 - 68902030177800*x^23 + 324474338021870*x^22 + 346500573850454*
x^21 - 4441518281684956*x^20 - 571126785242366*x^19 + 22736146340724250*x^18 - 100140780129284*x^17 - 63742636
856310672*x^16 + 1930227608742956*x^15 + 113272298306222934*x^14 - 3707342105075886*x^13 - 136316462990101068*
x^12 + 3707342105075886*x^11 + 113272298306222934*x^10 - 1930227608742956*x^9 - 63742636856310672*x^8 + 100140
780129284*x^7 + 22736146340724250*x^6 + 571126785242366*x^5 - 4441518281684956*x^4 - 346500573850454*x^3 + 324
474338021870*x^2 + sqrt(10)*(1008132874294*x^24 - 14888655494103*x^23 + 24952193442996*x^22 + 469321825791782*
x^21 - 1529467312445276*x^20 - 4322697049104696*x^19 + 14475801363390484*x^18 + 19894689930581267*x^17 - 59851
766151398456*x^16 - 52525242803693941*x^15 + 133501316828030084*x^14 + 84262553164312340*x^13 - 17326421378066
9712*x^12 - 84262553164312340*x^11 + 133501316828030084*x^10 + 52525242803693941*x^9 - 59851766151398456*x^8 -
 19894689930581267*x^7 + 14475801363390484*x^6 + 4322697049104696*x^5 - 1529467312445276*x^4 - 469321825791782
*x^3 + 24952193442996*x^2 + 14888655494103*x + 1008132874294) + 68902030177800*x + 3913944654568) + 1639244179
085383461600*x + 113784843418327559700)/(107276170508371881*x^24 - 2441300271537795968*x^23 + 1756930051626631
6746*x^22 - 16092110779172766528*x^21 - 288186279976289132707*x^20 + 672190931265926674240*x^19 + 194058290669
7052688958*x^18 - 4803572187578142625280*x^17 - 6971857565484310566550*x^16 + 15648066394863653981184*x^15 + 1
4595578320760563174802*x^14 - 27544211233989398727872*x^13 - 18571557939452786508651*x^12 + 275442112339893987
27872*x^11 + 14595578320760563174802*x^10 - 15648066394863653981184*x^9 - 6971857565484310566550*x^8 + 4803572
187578142625280*x^7 + 1940582906697052688958*x^6 - 672190931265926674240*x^5 - 288186279976289132707*x^4 + 160
92110779172766528*x^3 + 17569300516266316746*x^2 + 2441300271537795968*x + 107276170508371881)) - 1/100*10^(3/
4)*sqrt(5)*sqrt(2*sqrt(10) + 20)*arctan(-1/900*(113784843418327559700*x^24 - 1639244179085383461600*x^23 + 254
5884655734590980200*x^22 + 50821242952569317006400*x^21 - 158890131031597690275900*x^20 - 46505023743490864315
2000*x^19 + 1504931893222349028684600*x^18 + 2129607736390894049244000*x^17 - 6223660771133927367315000*x^16 -
 5602507922355181669879200*x^15 + 13883484083974894847147400*x^14 + 8971054092806414576313600*x^13 - 180190918
72612617624668700*x^12 - 8971054092806414576313600*x^11 + 13883484083974894847147400*x^10 + 560250792235518166
9879200*x^9 - 6223660771133927367315000*x^8 - 2129607736390894049244000*x^7 + 1504931893222349028684600*x^6 +
465050237434908643152000*x^5 - 158890131031597690275900*x^4 - 50821242952569317006400*x^3 + 254588465573459098
0200*x^2 - 17*sqrt(2)*((10^(3/4)*(sqrt(10)*sqrt(5)*(3353156233262299*x^24 - 49929701526135692*x^23 + 105488376
266631398*x^22 + 1245603618916311800*x^21 - 3729015017341775649*x^20 - 12574465648522249504*x^19 + 28773197469
313801650*x^18 + 61934497275400333884*x^17 - 104756202796838736914*x^16 - 169570606189065646500*x^15 + 2170840
68264548775966*x^14 + 276610136446380234320*x^13 - 274989879451094143625*x^12 - 276610136446380234320*x^11 + 2
17084068264548775966*x^10 + 169570606189065646500*x^9 - 104756202796838736914*x^8 - 61934497275400333884*x^7 +
 28773197469313801650*x^6 + 12574465648522249504*x^5 - 3729015017341775649*x^4 - 1245603618916311800*x^3 + 105
488376266631398*x^2 + 49929701526135692*x + 3353156233262299) + 10*sqrt(5)*(1603095924058903*x^24 - 2842494916
3733294*x^23 + 129882833582498276*x^22 + 237455573469526820*x^21 - 2351987676228007923*x^20 - 4789838754854734
48*x^19 + 15731365763207610420*x^18 - 1230955405989439242*x^17 - 54757871112862546478*x^16 + 75442913655498181
50*x^15 + 111831054251172443652*x^14 - 14992902143911637200*x^13 - 141157732738263834575*x^12 + 14992902143911
637200*x^11 + 111831054251172443652*x^10 - 7544291365549818150*x^9 - 54757871112862546478*x^8 + 12309554059894
39242*x^7 + 15731365763207610420*x^6 + 478983875485473448*x^5 - 2351987676228007923*x^4 - 237455573469526820*x
^3 + 129882833582498276*x^2 + 28424949163733294*x + 1603095924058903)) + 160*10^(1/4)*(2*sqrt(10)*sqrt(5)*(258
26047493168*x^24 - 423236377284370*x^23 + 1395910997963665*x^22 + 7977814049523724*x^21 - 39442227900912726*x^
20 - 55889144962836926*x^19 + 308168334219135075*x^18 + 191213417542285986*x^17 - 1166656659447404872*x^16 - 3
71314532640925644*x^15 + 2495683096606018509*x^14 + 476597108839388734*x^13 - 3197719822327310998*x^12 - 47659
7108839388734*x^11 + 2495683096606018509*x^10 + 371314532640925644*x^9 - 1166656659447404872*x^8 - 19121341754
2285986*x^7 + 308168334219135075*x^6 + 55889144962836926*x^5 - 39442227900912726*x^4 - 7977814049523724*x^3 +
1395910997963665*x^2 + 423236377284370*x + 25826047493168) + sqrt(5)*(167291197847878*x^24 - 2821779357503405*
x^23 + 10956936121559405*x^22 + 35249787921032759*x^21 - 204017758359407706*x^20 - 312436240830040741*x^19 + 1
399334424550385115*x^18 + 1735800349150666551*x^17 - 4987682548518497012*x^16 - 5368381759522587789*x^15 + 103
87543967266594209*x^14 + 9352242816540595199*x^13 - 13220291544952190258*x^12 - 9352242816540595199*x^11 + 103
87543967266594209*x^10 + 5368381759522587789*x^9 - 4987682548518497012*x^8 - 1735800349150666551*x^7 + 1399334
424550385115*x^6 + 312436240830040741*x^5 - 204017758359407706*x^4 - 35249787921032759*x^3 + 10956936121559405
*x^2 + 2821779357503405*x + 167291197847878)))*sqrt(2*sqrt(10) + 20) + 40*(7575923063334080*x^24 - 12084680003
6516320*x^23 + 365161535455038400*x^22 + 2264233628233167520*x^21 - 8547268899540342720*x^20 - 247226702746279
58240*x^19 + 64588595464717514880*x^18 + 139472668675247371680*x^17 - 248284848145133462080*x^16 - 42771300872
2866409440*x^15 + 544910109896139954240*x^14 + 743321346231612491680*x^13 - 707219317504256947840*x^12 - 74332
1346231612491680*x^11 + 544910109896139954240*x^10 + 427713008722866409440*x^9 - 248284848145133462080*x^8 - 1
39472668675247371680*x^7 + 64588595464717514880*x^6 + 24722670274627958240*x^5 - 8547268899540342720*x^4 - 226
4233628233167520*x^3 + 365161535455038400*x^2 + sqrt(10)*(3941811839020250*x^24 - 64348465775688550*x^23 + 225
885647210950450*x^22 + 941020417723504900*x^21 - 4570387865727921300*x^20 - 7999664382660162200*x^19 + 2958266
6125955145150*x^18 + 37508204562313810350*x^17 - 97777371150634079200*x^16 - 100481781887576998050*x^15 + 1919
89493413612300650*x^14 + 162041691927783863200*x^13 - 238833086260082653000*x^12 - 162041691927783863200*x^11
+ 191989493413612300650*x^10 + 100481781887576998050*x^9 - 97777371150634079200*x^8 - 37508204562313810350*x^7
 + 29582666125955145150*x^6 + 7999664382660162200*x^5 - 4570387865727921300*x^4 - 941020417723504900*x^3 + 225
885647210950450*x^2 + sqrt(10)*(1160866886007179*x^24 - 17845297455339433*x^23 + 44598386983244311*x^22 + 4074
27605373345862*x^21 - 1374776119098044166*x^20 - 4157698868742924356*x^19 + 10927521516291262449*x^18 + 216527
75358983721237*x^17 - 41131357142108902096*x^16 - 62264224681051657251*x^15 + 87175766672417705979*x^14 + 1042
83289775600539600*x^13 - 111301173832927037212*x^12 - 104283289775600539600*x^11 + 87175766672417705979*x^10 +
 62264224681051657251*x^9 - 41131357142108902096*x^8 - 21652775358983721237*x^7 + 10927521516291262449*x^6 + 4
157698868742924356*x^5 - 1374776119098044166*x^4 - 407427605373345862*x^3 + 44598386983244311*x^2 + 1784529745
5339433*x + 1160866886007179) + 64348465775688550*x + 3941811839020250) + 462400*sqrt(10)*(5159465558*x^24 - 8
1725010367*x^23 + 233461919938*x^22 + 1698386911873*x^21 - 6557645525970*x^20 - 15828018471263*x^19 + 48718561
372260*x^18 + 76722347870157*x^17 - 172301271560638*x^16 - 208531808984739*x^15 + 348922649378334*x^14 + 33814
0378140253*x^13 - 438000414906964*x^12 - 338140378140253*x^11 + 348922649378334*x^10 + 208531808984739*x^9 - 1
72301271560638*x^8 - 76722347870157*x^7 + 48718561372260*x^6 + 15828018471263*x^5 - 6557645525970*x^4 - 169838
6911873*x^3 + 233461919938*x^2 + 81725010367*x + 5159465558) + 120846800036516320*x + 7575923063334080)*sqrt((
2*x^2 - x - 2)/(x^2 + x - 1)))*sqrt((400*x^4 + 200*x^3 - 2*10^(1/4)*(sqrt(10)*sqrt(5)*(2*x^4 + x^3 - 5*x^2 - x
 + 2) + 5*sqrt(5)*(x^4 + 2*x^3 - x^2 - 2*x + 1))*sqrt(2*sqrt(10) + 20)*sqrt((2*x^2 - x - 2)/(x^2 + x - 1)) - 1
000*x^2 + 5*sqrt(10)*(17*x^4 + 4*x^3 - 29*x^2 - 4*x + 17) - 200*x + 400)/(x^4 - x^2 + 1)) - 86700*(10^(3/4)*(s
qrt(10)*sqrt(5)*(2442716712885*x^24 - 13728698401842*x^23 - 293186299957362*x^22 + 2310137367621964*x^21 + 145
1517102418989*x^20 - 33418412281395816*x^19 - 7178198464024966*x^18 + 183000238657163274*x^17 + 32979923853964
410*x^16 - 518819784198533318*x^15 - 84251682815179386*x^14 + 855052654520890752*x^13 + 114637825872128285*x^1
2 - 855052654520890752*x^11 - 84251682815179386*x^10 + 518819784198533318*x^9 + 32979923853964410*x^8 - 183000
238657163274*x^7 - 7178198464024966*x^6 + 33418412281395816*x^5 + 1451517102418989*x^4 - 2310137367621964*x^3
- 293186299957362*x^2 + 13728698401842*x + 2442716712885) + 2*sqrt(5)*(14773174320516*x^24 - 268375520761317*x
^23 + 1309307163746259*x^22 + 1740271771554118*x^21 - 22961098057094619*x^20 + 1995670027514196*x^19 + 1599953
61247050941*x^18 - 36026323622261007*x^17 - 580081963181949234*x^16 + 111975250170214681*x^15 + 12166562719087
09911*x^14 - 182294736937952040*x^13 - 1549896649785392773*x^12 + 182294736937952040*x^11 + 121665627190870991
1*x^10 - 111975250170214681*x^9 - 580081963181949234*x^8 + 36026323622261007*x^7 + 159995361247050941*x^6 - 19
95670027514196*x^5 - 22961098057094619*x^4 - 1740271771554118*x^3 + 1309307163746259*x^2 + 268375520761317*x +
 14773174320516)) + 32*10^(1/4)*(sqrt(10)*sqrt(5)*(353203013202*x^24 - 5252268148153*x^23 + 9599899316880*x^22
 + 157318487399903*x^21 - 528463660887158*x^20 - 1352806046560081*x^19 + 4754204159940742*x^18 + 5302733573770
307*x^17 - 19750563646089182*x^16 - 12045907326764941*x^15 + 44826810647943796*x^14 + 17772009847998347*x^13 -
 58660810041591536*x^12 - 17772009847998347*x^11 + 44826810647943796*x^10 + 12045907326764941*x^9 - 1975056364
6089182*x^8 - 5302733573770307*x^7 + 4754204159940742*x^6 + 1352806046560081*x^5 - 528463660887158*x^4 - 15731
8487399903*x^3 + 9599899316880*x^2 + 5252268148153*x + 353203013202) + 5*sqrt(5)*(239278797030*x^24 - 38941040
51149*x^23 + 14241545124129*x^22 + 43944733024733*x^21 - 190384021067588*x^20 - 581008343105785*x^19 + 9227663
99143729*x^18 + 4154684453315363*x^17 - 1978225802691326*x^16 - 14164309049142511*x^15 + 2165933895533713*x^14
 + 25503495102958511*x^13 - 1877306138593118*x^12 - 25503495102958511*x^11 + 2165933895533713*x^10 + 141643090
49142511*x^9 - 1978225802691326*x^8 - 4154684453315363*x^7 + 922766399143729*x^6 + 581008343105785*x^5 - 19038
4021067588*x^4 - 43944733024733*x^3 + 14241545124129*x^2 + 3894104051149*x + 239278797030)))*sqrt(2*sqrt(10) +
 20)*sqrt((2*x^2 - x - 2)/(x^2 + x - 1)) + 1300500*sqrt(10)*(33568388558203*x^24 - 575319820414644*x^23 + 2600
060853588106*x^22 + 3231189194065640*x^21 - 35687236548135733*x^20 - 7645624101059968*x^19 + 18267840254033115
0*x^18 + 9974894492537668*x^17 - 512039294651286218*x^16 - 7794844652266140*x^15 + 909771845956721522*x^14 + 3
856229278651280*x^13 - 1094808769706185685*x^12 - 3856229278651280*x^11 + 909771845956721522*x^10 + 7794844652
266140*x^9 - 512039294651286218*x^8 - 9974894492537668*x^7 + 182678402540331150*x^6 + 7645624101059968*x^5 - 3
5687236548135733*x^4 - 3231189194065640*x^3 + 2600060853588106*x^2 + 575319820414644*x + 33568388558203) - 104
04000*sqrt(10)*(3913944654568*x^24 - 68902030177800*x^23 + 324474338021870*x^22 + 346500573850454*x^21 - 44415
18281684956*x^20 - 571126785242366*x^19 + 22736146340724250*x^18 - 100140780129284*x^17 - 63742636856310672*x^
16 + 1930227608742956*x^15 + 113272298306222934*x^14 - 3707342105075886*x^13 - 136316462990101068*x^12 + 37073
42105075886*x^11 + 113272298306222934*x^10 - 1930227608742956*x^9 - 63742636856310672*x^8 + 100140780129284*x^
7 + 22736146340724250*x^6 + 571126785242366*x^5 - 4441518281684956*x^4 - 346500573850454*x^3 + 324474338021870
*x^2 + sqrt(10)*(1008132874294*x^24 - 14888655494103*x^23 + 24952193442996*x^22 + 469321825791782*x^21 - 15294
67312445276*x^20 - 4322697049104696*x^19 + 14475801363390484*x^18 + 19894689930581267*x^17 - 59851766151398456
*x^16 - 52525242803693941*x^15 + 133501316828030084*x^14 + 84262553164312340*x^13 - 173264213780669712*x^12 -
84262553164312340*x^11 + 133501316828030084*x^10 + 52525242803693941*x^9 - 59851766151398456*x^8 - 19894689930
581267*x^7 + 14475801363390484*x^6 + 4322697049104696*x^5 - 1529467312445276*x^4 - 469321825791782*x^3 + 24952
193442996*x^2 + 14888655494103*x + 1008132874294) + 68902030177800*x + 3913944654568) + 1639244179085383461600
*x + 113784843418327559700)/(107276170508371881*x^24 - 2441300271537795968*x^23 + 17569300516266316746*x^22 -
16092110779172766528*x^21 - 288186279976289132707*x^20 + 672190931265926674240*x^19 + 1940582906697052688958*x
^18 - 4803572187578142625280*x^17 - 6971857565484310566550*x^16 + 15648066394863653981184*x^15 + 1459557832076
0563174802*x^14 - 27544211233989398727872*x^13 - 18571557939452786508651*x^12 + 27544211233989398727872*x^11 +
 14595578320760563174802*x^10 - 15648066394863653981184*x^9 - 6971857565484310566550*x^8 + 4803572187578142625
280*x^7 + 1940582906697052688958*x^6 - 672190931265926674240*x^5 - 288186279976289132707*x^4 + 160921107791727
66528*x^3 + 17569300516266316746*x^2 + 2441300271537795968*x + 107276170508371881)) - 1/1200*10^(1/4)*(sqrt(10
)*sqrt(5) - 10*sqrt(5))*sqrt(2*sqrt(10) + 20)*log(14450*(400*x^4 + 200*x^3 + 2*10^(1/4)*(sqrt(10)*sqrt(5)*(2*x
^4 + x^3 - 5*x^2 - x + 2) + 5*sqrt(5)*(x^4 + 2*x^3 - x^2 - 2*x + 1))*sqrt(2*sqrt(10) + 20)*sqrt((2*x^2 - x - 2
)/(x^2 + x - 1)) - 1000*x^2 + 5*sqrt(10)*(17*x^4 + 4*x^3 - 29*x^2 - 4*x + 17) - 200*x + 400)/(x^4 - x^2 + 1))
+ 1/1200*10^(1/4)*(sqrt(10)*sqrt(5) - 10*sqrt(5))*sqrt(2*sqrt(10) + 20)*log(14450*(400*x^4 + 200*x^3 - 2*10^(1
/4)*(sqrt(10)*sqrt(5)*(2*x^4 + x^3 - 5*x^2 - x + 2) + 5*sqrt(5)*(x^4 + 2*x^3 - x^2 - 2*x + 1))*sqrt(2*sqrt(10)
 + 20)*sqrt((2*x^2 - x - 2)/(x^2 + x - 1)) - 1000*x^2 + 5*sqrt(10)*(17*x^4 + 4*x^3 - 29*x^2 - 4*x + 17) - 200*
x + 400)/(x^4 - x^2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/((2*x^2-x-2)/(x^2+x-1))^(1/2)/(x^4-x^2+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 4.24, size = 841, normalized size = 8.49

method result size
trager \(\frac {\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \ln \left (-\frac {-1620 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{4} x +260 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+459 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}+260 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x -207 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x -260 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+265 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, x^{2}-459 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right )+51 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}+265 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, x -3 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x -265 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}-51 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+10\right )}{3 x^{2}+20 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +x -3}\right )}{10}-\RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-16200 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{5} x +260 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}-4590 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{2}+260 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x -1170 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x -260 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}-239 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, x^{2}+4590 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3}+51 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{2}-239 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}\, x +15 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x +239 \sqrt {-\frac {-2 x^{2}+x +2}{x^{2}+x -1}}-51 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{20 \RootOf \left (40 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x -3 x^{2}+x +3}\right )\) \(841\)
default \(\text {Expression too large to display}\) \(7672\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10)*ln(-(-1620*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10)*
RootOf(40*_Z^4+4*_Z^2+1)^4*x+260*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*RootOf(40*_Z^4+4*_Z^2+1)^2*x^2+459*RootOf(40*
_Z^4+4*_Z^2+1)^2*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10)*x^2+260*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*RootOf
(40*_Z^4+4*_Z^2+1)^2*x-207*RootOf(40*_Z^4+4*_Z^2+1)^2*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10)*x-260*(-(
-2*x^2+x+2)/(x^2+x-1))^(1/2)*RootOf(40*_Z^4+4*_Z^2+1)^2+265*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*x^2-459*RootOf(40*
_Z^4+4*_Z^2+1)^2*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10)+51*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+
10)*x^2+265*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*x-3*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10)*x-265*(-(-2*x^2
+x+2)/(x^2+x-1))^(1/2)-51*RootOf(_Z^2+100*RootOf(40*_Z^4+4*_Z^2+1)^2+10))/(3*x^2+20*RootOf(40*_Z^4+4*_Z^2+1)^2
*x+x-3))-RootOf(40*_Z^4+4*_Z^2+1)*ln((-16200*RootOf(40*_Z^4+4*_Z^2+1)^5*x+260*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*
RootOf(40*_Z^4+4*_Z^2+1)^2*x^2-4590*RootOf(40*_Z^4+4*_Z^2+1)^3*x^2+260*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*RootOf(
40*_Z^4+4*_Z^2+1)^2*x-1170*RootOf(40*_Z^4+4*_Z^2+1)^3*x-260*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*RootOf(40*_Z^4+4*_
Z^2+1)^2-239*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*x^2+4590*RootOf(40*_Z^4+4*_Z^2+1)^3+51*RootOf(40*_Z^4+4*_Z^2+1)*x
^2-239*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)*x+15*RootOf(40*_Z^4+4*_Z^2+1)*x+239*(-(-2*x^2+x+2)/(x^2+x-1))^(1/2)-51*
RootOf(40*_Z^4+4*_Z^2+1))/(20*RootOf(40*_Z^4+4*_Z^2+1)^2*x-3*x^2+x+3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+1}{\sqrt {-\frac {-2\,x^2+x+2}{x^2+x-1}}\,\left (x^4-x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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**2-x-2)/(x**2+x-1))**(1/2)/(x**4-x**2+1),x)

[Out]

Timed out

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