3.28.40 \(\int \frac {(-1+x^2) \sqrt {1-x-x^2+x^3+x^4}}{x^2 (1+x^2)} \, dx\)
Optimal. Leaf size=254 \[ \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3+62 \text {$\#$1}^2+12 \text {$\#$1}+9\& ,\frac {3 \text {$\#$1}^2 \log \left (\text {$\#$1} x-2 x^2+2 \sqrt {x^4+x^3-x^2-x+1}-x+2\right )-3 \text {$\#$1}^2 \log (x)+10 \text {$\#$1} \log \left (\text {$\#$1} x-2 x^2+2 \sqrt {x^4+x^3-x^2-x+1}-x+2\right )-9 \log \left (\text {$\#$1} x-2 x^2+2 \sqrt {x^4+x^3-x^2-x+1}-x+2\right )-10 \text {$\#$1} \log (x)+9 \log (x)}{\text {$\#$1}^3-3 \text {$\#$1}^2+31 \text {$\#$1}+3}\& \right ]+\frac {\sqrt {x^4+x^3-x^2-x+1}}{x}-\frac {1}{2} \log \left (-2 x^2+2 \sqrt {x^4+x^3-x^2-x+1}-x+2\right )+\frac {\log (x)}{2} \]
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Rubi [F] time = 0.55, 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 ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((-1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4])/(x^2*(1 + x^2)),x]
[Out]
I*Defer[Int][Sqrt[1 - x - x^2 + x^3 + x^4]/(I - x), x] - Defer[Int][Sqrt[1 - x - x^2 + x^3 + x^4]/x^2, x] + I*
Defer[Int][Sqrt[1 - x - x^2 + x^3 + x^4]/(I + x), x]
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx &=\int \left (-\frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2}+\frac {2 \sqrt {1-x-x^2+x^3+x^4}}{1+x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt {1-x-x^2+x^3+x^4}}{1+x^2} \, dx-\int \frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2} \, dx\\ &=2 \int \left (\frac {i \sqrt {1-x-x^2+x^3+x^4}}{2 (i-x)}+\frac {i \sqrt {1-x-x^2+x^3+x^4}}{2 (i+x)}\right ) \, dx-\int \frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2} \, dx\\ &=i \int \frac {\sqrt {1-x-x^2+x^3+x^4}}{i-x} \, dx+i \int \frac {\sqrt {1-x-x^2+x^3+x^4}}{i+x} \, dx-\int \frac {\sqrt {1-x-x^2+x^3+x^4}}{x^2} \, dx\\ \end {align*}
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Mathematica [C] time = 6.15, size = 8085, normalized size = 31.83 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[((-1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4])/(x^2*(1 + x^2)),x]
[Out]
Result too large to show
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IntegrateAlgebraic [A] time = 0.00, size = 232, normalized size = 0.91 \begin {gather*} \frac {\sqrt {1-x-x^2+x^3+x^4}}{x}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )-\frac {1}{2} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log (x)+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}-8 \log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4])/(x^2*(1 + x^2)),x]
[Out]
Sqrt[1 - x - x^2 + x^3 + x^4]/x + Log[x]/2 - Log[2 - x - 2*x^2 + 2*Sqrt[1 - x - x^2 + x^3 + x^4]]/2 - RootSum[
5 - 16*#1 + 14*#1^2 + #1^4 & , (-Log[x] + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] + 8*Log[x]*#1 -
8*Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1 - 3*Log[x]*#1^2 + 3*Log[1 - x^2 + Sqrt[1 - x - x^2 +
x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/2
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fricas [B] time = 5.72, size = 4897, normalized size = 19.28
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)*(x^4+x^3-x^2-x+1)^(1/2)/x^2/(x^2+1),x, algorithm="fricas")
[Out]
-1/208*(8*13^(3/4)*sqrt(2)*x*sqrt(3*sqrt(13) + 13)*arctan(-1/78*(9973402820649333*x^24 + 22127143477092732*x^2
3 + 411425006050475352*x^22 - 2674034968526173980*x^21 - 8817595222945016430*x^20 + 26775269173700401068*x^19
+ 78094051238357921208*x^18 - 79698905838822826764*x^17 - 288484945304365905381*x^16 + 103539974542743456120*x
^15 + 564311426035174966512*x^14 - 89471281890239169336*x^13 - 696743417151333048900*x^12 + 894712818902391693
36*x^11 + 564311426035174966512*x^10 - 103539974542743456120*x^9 - 288484945304365905381*x^8 + 796989058388228
26764*x^7 + 78094051238357921208*x^6 - 26775269173700401068*x^5 - 8817595222945016430*x^4 + 267403496852617398
0*x^3 + 411425006050475352*x^2 + 975*sqrt(x^4 + x^3 - x^2 - x + 1)*(13^(3/4)*(sqrt(13)*sqrt(2)*(35341242867*x^
22 + 1855733013342*x^21 - 13065830234977*x^20 - 13707953161500*x^19 + 161237193602453*x^18 + 131938117769830*x
^17 - 841354492514479*x^16 - 943610658660240*x^15 + 1815156449248878*x^14 + 2328772665064508*x^13 - 2454398966
001146*x^12 - 3055785341153960*x^11 + 2454398966001146*x^10 + 2328772665064508*x^9 - 1815156449248878*x^8 - 94
3610658660240*x^7 + 841354492514479*x^6 + 131938117769830*x^5 - 161237193602453*x^4 - 13707953161500*x^3 + 130
65830234977*x^2 + 1855733013342*x - 35341242867) + 13*sqrt(2)*(27984641913*x^22 - 681850618938*x^21 + 18324707
32861*x^20 + 10480832872820*x^19 - 25798153427169*x^18 - 87720458521554*x^17 + 119868804550803*x^16 + 40980739
0513968*x^15 - 171761161217974*x^14 - 885623303420404*x^13 + 139243488716306*x^12 + 1121192318939832*x^11 - 13
9243488716306*x^10 - 885623303420404*x^9 + 171761161217974*x^8 + 409807390513968*x^7 - 119868804550803*x^6 - 8
7720458521554*x^5 + 25798153427169*x^4 + 10480832872820*x^3 - 1832470732861*x^2 - 681850618938*x - 27984641913
)) + 52*13^(1/4)*(sqrt(13)*sqrt(2)*(4145451651*x^22 + 22790133663*x^21 - 787396271270*x^20 + 3268260459922*x^1
9 + 2127235898078*x^18 - 26388586645821*x^17 + 4534478469577*x^16 + 96402875532632*x^15 - 39927895710530*x^14
- 219325332502114*x^13 + 73075426941692*x^12 + 286730467695148*x^11 - 73075426941692*x^10 - 219325332502114*x^
9 + 39927895710530*x^8 + 96402875532632*x^7 - 4534478469577*x^6 - 26388586645821*x^5 - 2127235898078*x^4 + 326
8260459922*x^3 + 787396271270*x^2 + 22790133663*x - 4145451651) + sqrt(2)*(19518507459*x^22 - 258433982565*x^2
1 + 1268510129270*x^20 - 4641107948794*x^19 + 7873387874194*x^18 + 42079083144127*x^17 - 120347500997751*x^16
- 207088711742488*x^15 + 446369667432350*x^14 + 582726984044198*x^13 - 747970503528716*x^12 - 801333993329116*
x^11 + 747970503528716*x^10 + 582726984044198*x^9 - 446369667432350*x^8 - 207088711742488*x^7 + 12034750099775
1*x^6 + 42079083144127*x^5 - 7873387874194*x^4 - 4641107948794*x^3 - 1268510129270*x^2 - 258433982565*x - 1951
8507459)))*sqrt(3*sqrt(13) + 13) - 5*sqrt(13)*(2*(419409782458524*x^22 - 734551553772036*x^21 - 18129135798825
480*x^20 + 46342415724970040*x^19 + 147660832091121960*x^18 - 281814250716297780*x^17 - 771997052407149868*x^1
6 + 581202262545036704*x^15 + 1981992402699630360*x^14 - 560878898811974600*x^13 - 2949154913394612208*x^12 +
508183375662515280*x^11 + 2949154913394612208*x^10 - 560878898811974600*x^9 - 1981992402699630360*x^8 + 581202
262545036704*x^7 + 771997052407149868*x^6 - 281814250716297780*x^5 - 147660832091121960*x^4 + 4634241572497004
0*x^3 + 18129135798825480*x^2 + sqrt(13)*(114360666044625*x^22 - 82159206468300*x^21 - 5119203489735075*x^20 +
9000368541541000*x^19 + 50566728981273375*x^18 - 68694538787756700*x^17 - 265377087786179525*x^16 + 188115112
145740000*x^15 + 743684297270684250*x^14 - 260846258391823000*x^13 - 1192582070334467150*x^12 + 27349548522511
8000*x^11 + 1192582070334467150*x^10 - 260846258391823000*x^9 - 743684297270684250*x^8 + 188115112145740000*x^
7 + 265377087786179525*x^6 - 68694538787756700*x^5 - 50566728981273375*x^4 + 9000368541541000*x^3 + 5119203489
735075*x^2 + sqrt(13)*(32174649863631*x^22 - 54667700989668*x^21 - 1161454885034925*x^20 + 2231591213841080*x^
19 + 11317460228134065*x^18 - 15147501756126036*x^17 - 57433522492693531*x^16 + 36165933376158752*x^15 + 14904
6541912200870*x^14 - 51483285976208840*x^13 - 232497595537749778*x^12 + 55228511827706448*x^11 + 2324975955377
49778*x^10 - 51483285976208840*x^9 - 149046541912200870*x^8 + 36165933376158752*x^7 + 57433522492693531*x^6 -
15147501756126036*x^5 - 11317460228134065*x^4 + 2231591213841080*x^3 + 1161454885034925*x^2 - 54667700989668*x
- 32174649863631) - 82159206468300*x - 114360666044625) + 16900*sqrt(13)*(6910217919*x^22 - 14188512393*x^21
- 264938196402*x^20 + 560173342750*x^19 + 2885410276890*x^18 - 4195778936853*x^17 - 16049843908931*x^16 + 1065
9197774440*x^15 + 43981621047990*x^14 - 15468175661410*x^13 - 70387829519228*x^12 + 16724046063348*x^11 + 7038
7829519228*x^10 - 15468175661410*x^9 - 43981621047990*x^8 + 10659197774440*x^7 + 16049843908931*x^6 - 41957789
36853*x^5 - 2885410276890*x^4 + 560173342750*x^3 + 264938196402*x^2 - 14188512393*x - 6910217919) - 7345515537
72036*x - 419409782458524)*sqrt(x^4 + x^3 - x^2 - x + 1) + (13^(3/4)*(sqrt(13)*sqrt(2)*(5064601110759*x^24 - 1
5935521225248*x^23 - 145953298037646*x^22 + 442342162318472*x^21 + 1509242233362130*x^20 - 3824593137929864*x^
19 - 9272490241084918*x^18 + 15231739690849728*x^17 + 35123888634951257*x^16 - 29158744370317408*x^15 - 721218
52990916732*x^14 + 37093947378248880*x^13 + 90527094861258716*x^12 - 37093947378248880*x^11 - 7212185299091673
2*x^10 + 29158744370317408*x^9 + 35123888634951257*x^8 - 15231739690849728*x^7 - 9272490241084918*x^6 + 382459
3137929864*x^5 + 1509242233362130*x^4 - 442342162318472*x^3 - 145953298037646*x^2 + 15935521225248*x + 5064601
110759) + 2*sqrt(2)*(8344585217061*x^24 + 22313480635683*x^23 - 454941103887909*x^22 - 29015155993837*x^21 + 5
263571461584220*x^20 + 1350672090063169*x^19 - 28025155569909697*x^18 - 17323853348703663*x^17 + 7170945240527
6003*x^16 + 53329447307333918*x^15 - 117861135465169178*x^14 - 87340455501519330*x^13 + 137540784460766264*x^1
2 + 87340455501519330*x^11 - 117861135465169178*x^10 - 53329447307333918*x^9 + 71709452405276003*x^8 + 1732385
3348703663*x^7 - 28025155569909697*x^6 - 1350672090063169*x^5 + 5263571461584220*x^4 + 29015155993837*x^3 - 45
4941103887909*x^2 - 22313480635683*x + 8344585217061)) + 2*13^(1/4)*(sqrt(13)*sqrt(2)*(8950902551613*x^24 - 17
420951995029*x^23 - 280276398361365*x^22 + 267465288044777*x^21 + 3726162631678360*x^20 - 556533136788845*x^19
- 23463111563130721*x^18 - 10788785991868263*x^17 + 71350869932571899*x^16 + 52559233428319022*x^15 - 1216571
74574889866*x^14 - 96889156840890966*x^13 + 143212893636486752*x^12 + 96889156840890966*x^11 - 121657174574889
866*x^10 - 52559233428319022*x^9 + 71350869932571899*x^8 + 10788785991868263*x^7 - 23463111563130721*x^6 + 556
533136788845*x^5 + 3726162631678360*x^4 - 267465288044777*x^3 - 280276398361365*x^2 + 17420951995029*x + 89509
02551613) + 13*sqrt(2)*(2421742919211*x^24 - 354297570363*x^23 - 128501377823955*x^22 + 245465593952119*x^21 +
1223894974340720*x^20 - 1990800605349715*x^19 - 6408712012034087*x^18 + 6584370631518039*x^17 + 1837484873443
7053*x^16 - 14112688088835566*x^15 - 34459812164567302*x^14 + 19949971913218998*x^13 + 42245657417848144*x^12
- 19949971913218998*x^11 - 34459812164567302*x^10 + 14112688088835566*x^9 + 18374848734437053*x^8 - 6584370631
518039*x^7 - 6408712012034087*x^6 + 1990800605349715*x^5 + 1223894974340720*x^4 - 245465593952119*x^3 - 128501
377823955*x^2 + 354297570363*x + 2421742919211)))*sqrt(3*sqrt(13) + 13))*sqrt((52*x^4 + 52*x^3 - 13^(1/4)*sqrt
(x^4 + x^3 - x^2 - x + 1)*(sqrt(13)*sqrt(2)*(x^2 + 2*x - 1) + sqrt(2)*(5*x^2 - 2*x - 5))*sqrt(3*sqrt(13) + 13)
- 52*x^2 + sqrt(13)*(17*x^4 + 20*x^3 - 26*x^2 - 20*x + 17) - 52*x + 52)/(x^4 + 2*x^2 + 1)) + 12675*sqrt(13)*(
177838764957*x^24 + 3282751894572*x^23 - 12363069681816*x^22 - 47326717811724*x^21 + 72293202696770*x^20 + 288
662813058588*x^19 + 55148659933960*x^18 - 345299438215612*x^17 - 172702607180557*x^16 - 87498335053544*x^15 -
32309816748656*x^14 + 597073385926952*x^13 + 221414678444636*x^12 - 597073385926952*x^11 - 32309816748656*x^10
+ 87498335053544*x^9 - 172702607180557*x^8 + 345299438215612*x^7 + 55148659933960*x^6 - 288662813058588*x^5 +
72293202696770*x^4 + 47326717811724*x^3 - 12363069681816*x^2 - 3282751894572*x + 177838764957) + 50700*sqrt(1
3)*(67029569739*x^24 - 597586574235*x^23 - 446407157383*x^22 + 8646734742891*x^21 + 6344564056940*x^20 - 42154
169717695*x^19 - 55151396492523*x^18 + 27807985037327*x^17 + 50679508980909*x^16 + 24831639280746*x^15 + 52238
368574322*x^14 - 54374836791402*x^13 - 120901075366344*x^12 + 54374836791402*x^11 + 52238368574322*x^10 - 2483
1639280746*x^9 + 50679508980909*x^8 - 27807985037327*x^7 - 55151396492523*x^6 + 42154169717695*x^5 + 634456405
6940*x^4 - 8646734742891*x^3 - 446407157383*x^2 + sqrt(13)*(15948545679*x^24 + 18290888373*x^23 - 158782078101
5*x^22 + 3527145459591*x^21 + 19118859600980*x^20 - 33894946115435*x^19 - 125610923441243*x^18 + 1097475571029
11*x^17 + 436323477217897*x^16 - 158309885652374*x^15 - 848740127958478*x^14 + 153014850561262*x^13 + 10470256
75744616*x^12 - 153014850561262*x^11 - 848740127958478*x^10 + 158309885652374*x^9 + 436323477217897*x^8 - 1097
47557102911*x^7 - 125610923441243*x^6 + 33894946115435*x^5 + 19118859600980*x^4 - 3527145459591*x^3 - 15878207
81015*x^2 - 18290888373*x + 15948545679) + 597586574235*x + 67029569739) - 22127143477092732*x + 9973402820649
333)/(33216827121477*x^24 - 2060726880350592*x^23 + 1431075537830988*x^22 + 83828069684567680*x^21 - 145619826
687334470*x^20 - 798797730254046208*x^19 + 1013303893827450652*x^18 + 4124312348255593984*x^17 - 2350198326509
186389*x^16 - 10945125280902753920*x^15 + 2837136293429528728*x^14 + 17083854766580127616*x^13 - 2801179578979
740500*x^12 - 17083854766580127616*x^11 + 2837136293429528728*x^10 + 10945125280902753920*x^9 - 23501983265091
86389*x^8 - 4124312348255593984*x^7 + 1013303893827450652*x^6 + 798797730254046208*x^5 - 145619826687334470*x^
4 - 83828069684567680*x^3 + 1431075537830988*x^2 + 2060726880350592*x + 33216827121477)) + 8*13^(3/4)*sqrt(2)*
x*sqrt(3*sqrt(13) + 13)*arctan(1/78*(9973402820649333*x^24 + 22127143477092732*x^23 + 411425006050475352*x^22
- 2674034968526173980*x^21 - 8817595222945016430*x^20 + 26775269173700401068*x^19 + 78094051238357921208*x^18
- 79698905838822826764*x^17 - 288484945304365905381*x^16 + 103539974542743456120*x^15 + 564311426035174966512*
x^14 - 89471281890239169336*x^13 - 696743417151333048900*x^12 + 89471281890239169336*x^11 + 564311426035174966
512*x^10 - 103539974542743456120*x^9 - 288484945304365905381*x^8 + 79698905838822826764*x^7 + 7809405123835792
1208*x^6 - 26775269173700401068*x^5 - 8817595222945016430*x^4 + 2674034968526173980*x^3 + 411425006050475352*x
^2 - 975*sqrt(x^4 + x^3 - x^2 - x + 1)*(13^(3/4)*(sqrt(13)*sqrt(2)*(35341242867*x^22 + 1855733013342*x^21 - 13
065830234977*x^20 - 13707953161500*x^19 + 161237193602453*x^18 + 131938117769830*x^17 - 841354492514479*x^16 -
943610658660240*x^15 + 1815156449248878*x^14 + 2328772665064508*x^13 - 2454398966001146*x^12 - 30557853411539
60*x^11 + 2454398966001146*x^10 + 2328772665064508*x^9 - 1815156449248878*x^8 - 943610658660240*x^7 + 84135449
2514479*x^6 + 131938117769830*x^5 - 161237193602453*x^4 - 13707953161500*x^3 + 13065830234977*x^2 + 1855733013
342*x - 35341242867) + 13*sqrt(2)*(27984641913*x^22 - 681850618938*x^21 + 1832470732861*x^20 + 10480832872820*
x^19 - 25798153427169*x^18 - 87720458521554*x^17 + 119868804550803*x^16 + 409807390513968*x^15 - 1717611612179
74*x^14 - 885623303420404*x^13 + 139243488716306*x^12 + 1121192318939832*x^11 - 139243488716306*x^10 - 8856233
03420404*x^9 + 171761161217974*x^8 + 409807390513968*x^7 - 119868804550803*x^6 - 87720458521554*x^5 + 25798153
427169*x^4 + 10480832872820*x^3 - 1832470732861*x^2 - 681850618938*x - 27984641913)) + 52*13^(1/4)*(sqrt(13)*s
qrt(2)*(4145451651*x^22 + 22790133663*x^21 - 787396271270*x^20 + 3268260459922*x^19 + 2127235898078*x^18 - 263
88586645821*x^17 + 4534478469577*x^16 + 96402875532632*x^15 - 39927895710530*x^14 - 219325332502114*x^13 + 730
75426941692*x^12 + 286730467695148*x^11 - 73075426941692*x^10 - 219325332502114*x^9 + 39927895710530*x^8 + 964
02875532632*x^7 - 4534478469577*x^6 - 26388586645821*x^5 - 2127235898078*x^4 + 3268260459922*x^3 + 78739627127
0*x^2 + 22790133663*x - 4145451651) + sqrt(2)*(19518507459*x^22 - 258433982565*x^21 + 1268510129270*x^20 - 464
1107948794*x^19 + 7873387874194*x^18 + 42079083144127*x^17 - 120347500997751*x^16 - 207088711742488*x^15 + 446
369667432350*x^14 + 582726984044198*x^13 - 747970503528716*x^12 - 801333993329116*x^11 + 747970503528716*x^10
+ 582726984044198*x^9 - 446369667432350*x^8 - 207088711742488*x^7 + 120347500997751*x^6 + 42079083144127*x^5 -
7873387874194*x^4 - 4641107948794*x^3 - 1268510129270*x^2 - 258433982565*x - 19518507459)))*sqrt(3*sqrt(13) +
13) - 5*sqrt(13)*(2*(419409782458524*x^22 - 734551553772036*x^21 - 18129135798825480*x^20 + 46342415724970040
*x^19 + 147660832091121960*x^18 - 281814250716297780*x^17 - 771997052407149868*x^16 + 581202262545036704*x^15
+ 1981992402699630360*x^14 - 560878898811974600*x^13 - 2949154913394612208*x^12 + 508183375662515280*x^11 + 29
49154913394612208*x^10 - 560878898811974600*x^9 - 1981992402699630360*x^8 + 581202262545036704*x^7 + 771997052
407149868*x^6 - 281814250716297780*x^5 - 147660832091121960*x^4 + 46342415724970040*x^3 + 18129135798825480*x^
2 + sqrt(13)*(114360666044625*x^22 - 82159206468300*x^21 - 5119203489735075*x^20 + 9000368541541000*x^19 + 505
66728981273375*x^18 - 68694538787756700*x^17 - 265377087786179525*x^16 + 188115112145740000*x^15 + 74368429727
0684250*x^14 - 260846258391823000*x^13 - 1192582070334467150*x^12 + 273495485225118000*x^11 + 1192582070334467
150*x^10 - 260846258391823000*x^9 - 743684297270684250*x^8 + 188115112145740000*x^7 + 265377087786179525*x^6 -
68694538787756700*x^5 - 50566728981273375*x^4 + 9000368541541000*x^3 + 5119203489735075*x^2 + sqrt(13)*(32174
649863631*x^22 - 54667700989668*x^21 - 1161454885034925*x^20 + 2231591213841080*x^19 + 11317460228134065*x^18
- 15147501756126036*x^17 - 57433522492693531*x^16 + 36165933376158752*x^15 + 149046541912200870*x^14 - 5148328
5976208840*x^13 - 232497595537749778*x^12 + 55228511827706448*x^11 + 232497595537749778*x^10 - 514832859762088
40*x^9 - 149046541912200870*x^8 + 36165933376158752*x^7 + 57433522492693531*x^6 - 15147501756126036*x^5 - 1131
7460228134065*x^4 + 2231591213841080*x^3 + 1161454885034925*x^2 - 54667700989668*x - 32174649863631) - 8215920
6468300*x - 114360666044625) + 16900*sqrt(13)*(6910217919*x^22 - 14188512393*x^21 - 264938196402*x^20 + 560173
342750*x^19 + 2885410276890*x^18 - 4195778936853*x^17 - 16049843908931*x^16 + 10659197774440*x^15 + 4398162104
7990*x^14 - 15468175661410*x^13 - 70387829519228*x^12 + 16724046063348*x^11 + 70387829519228*x^10 - 1546817566
1410*x^9 - 43981621047990*x^8 + 10659197774440*x^7 + 16049843908931*x^6 - 4195778936853*x^5 - 2885410276890*x^
4 + 560173342750*x^3 + 264938196402*x^2 - 14188512393*x - 6910217919) - 734551553772036*x - 419409782458524)*s
qrt(x^4 + x^3 - x^2 - x + 1) - (13^(3/4)*(sqrt(13)*sqrt(2)*(5064601110759*x^24 - 15935521225248*x^23 - 1459532
98037646*x^22 + 442342162318472*x^21 + 1509242233362130*x^20 - 3824593137929864*x^19 - 9272490241084918*x^18 +
15231739690849728*x^17 + 35123888634951257*x^16 - 29158744370317408*x^15 - 72121852990916732*x^14 + 370939473
78248880*x^13 + 90527094861258716*x^12 - 37093947378248880*x^11 - 72121852990916732*x^10 + 29158744370317408*x
^9 + 35123888634951257*x^8 - 15231739690849728*x^7 - 9272490241084918*x^6 + 3824593137929864*x^5 + 15092422333
62130*x^4 - 442342162318472*x^3 - 145953298037646*x^2 + 15935521225248*x + 5064601110759) + 2*sqrt(2)*(8344585
217061*x^24 + 22313480635683*x^23 - 454941103887909*x^22 - 29015155993837*x^21 + 5263571461584220*x^20 + 13506
72090063169*x^19 - 28025155569909697*x^18 - 17323853348703663*x^17 + 71709452405276003*x^16 + 5332944730733391
8*x^15 - 117861135465169178*x^14 - 87340455501519330*x^13 + 137540784460766264*x^12 + 87340455501519330*x^11 -
117861135465169178*x^10 - 53329447307333918*x^9 + 71709452405276003*x^8 + 17323853348703663*x^7 - 28025155569
909697*x^6 - 1350672090063169*x^5 + 5263571461584220*x^4 + 29015155993837*x^3 - 454941103887909*x^2 - 22313480
635683*x + 8344585217061)) + 2*13^(1/4)*(sqrt(13)*sqrt(2)*(8950902551613*x^24 - 17420951995029*x^23 - 28027639
8361365*x^22 + 267465288044777*x^21 + 3726162631678360*x^20 - 556533136788845*x^19 - 23463111563130721*x^18 -
10788785991868263*x^17 + 71350869932571899*x^16 + 52559233428319022*x^15 - 121657174574889866*x^14 - 968891568
40890966*x^13 + 143212893636486752*x^12 + 96889156840890966*x^11 - 121657174574889866*x^10 - 52559233428319022
*x^9 + 71350869932571899*x^8 + 10788785991868263*x^7 - 23463111563130721*x^6 + 556533136788845*x^5 + 372616263
1678360*x^4 - 267465288044777*x^3 - 280276398361365*x^2 + 17420951995029*x + 8950902551613) + 13*sqrt(2)*(2421
742919211*x^24 - 354297570363*x^23 - 128501377823955*x^22 + 245465593952119*x^21 + 1223894974340720*x^20 - 199
0800605349715*x^19 - 6408712012034087*x^18 + 6584370631518039*x^17 + 18374848734437053*x^16 - 1411268808883556
6*x^15 - 34459812164567302*x^14 + 19949971913218998*x^13 + 42245657417848144*x^12 - 19949971913218998*x^11 - 3
4459812164567302*x^10 + 14112688088835566*x^9 + 18374848734437053*x^8 - 6584370631518039*x^7 - 640871201203408
7*x^6 + 1990800605349715*x^5 + 1223894974340720*x^4 - 245465593952119*x^3 - 128501377823955*x^2 + 354297570363
*x + 2421742919211)))*sqrt(3*sqrt(13) + 13))*sqrt((52*x^4 + 52*x^3 + 13^(1/4)*sqrt(x^4 + x^3 - x^2 - x + 1)*(s
qrt(13)*sqrt(2)*(x^2 + 2*x - 1) + sqrt(2)*(5*x^2 - 2*x - 5))*sqrt(3*sqrt(13) + 13) - 52*x^2 + sqrt(13)*(17*x^4
+ 20*x^3 - 26*x^2 - 20*x + 17) - 52*x + 52)/(x^4 + 2*x^2 + 1)) + 12675*sqrt(13)*(177838764957*x^24 + 32827518
94572*x^23 - 12363069681816*x^22 - 47326717811724*x^21 + 72293202696770*x^20 + 288662813058588*x^19 + 55148659
933960*x^18 - 345299438215612*x^17 - 172702607180557*x^16 - 87498335053544*x^15 - 32309816748656*x^14 + 597073
385926952*x^13 + 221414678444636*x^12 - 597073385926952*x^11 - 32309816748656*x^10 + 87498335053544*x^9 - 1727
02607180557*x^8 + 345299438215612*x^7 + 55148659933960*x^6 - 288662813058588*x^5 + 72293202696770*x^4 + 473267
17811724*x^3 - 12363069681816*x^2 - 3282751894572*x + 177838764957) + 50700*sqrt(13)*(67029569739*x^24 - 59758
6574235*x^23 - 446407157383*x^22 + 8646734742891*x^21 + 6344564056940*x^20 - 42154169717695*x^19 - 55151396492
523*x^18 + 27807985037327*x^17 + 50679508980909*x^16 + 24831639280746*x^15 + 52238368574322*x^14 - 54374836791
402*x^13 - 120901075366344*x^12 + 54374836791402*x^11 + 52238368574322*x^10 - 24831639280746*x^9 + 50679508980
909*x^8 - 27807985037327*x^7 - 55151396492523*x^6 + 42154169717695*x^5 + 6344564056940*x^4 - 8646734742891*x^3
- 446407157383*x^2 + sqrt(13)*(15948545679*x^24 + 18290888373*x^23 - 1587820781015*x^22 + 3527145459591*x^21
+ 19118859600980*x^20 - 33894946115435*x^19 - 125610923441243*x^18 + 109747557102911*x^17 + 436323477217897*x^
16 - 158309885652374*x^15 - 848740127958478*x^14 + 153014850561262*x^13 + 1047025675744616*x^12 - 153014850561
262*x^11 - 848740127958478*x^10 + 158309885652374*x^9 + 436323477217897*x^8 - 109747557102911*x^7 - 1256109234
41243*x^6 + 33894946115435*x^5 + 19118859600980*x^4 - 3527145459591*x^3 - 1587820781015*x^2 - 18290888373*x +
15948545679) + 597586574235*x + 67029569739) - 22127143477092732*x + 9973402820649333)/(33216827121477*x^24 -
2060726880350592*x^23 + 1431075537830988*x^22 + 83828069684567680*x^21 - 145619826687334470*x^20 - 79879773025
4046208*x^19 + 1013303893827450652*x^18 + 4124312348255593984*x^17 - 2350198326509186389*x^16 - 10945125280902
753920*x^15 + 2837136293429528728*x^14 + 17083854766580127616*x^13 - 2801179578979740500*x^12 - 17083854766580
127616*x^11 + 2837136293429528728*x^10 + 10945125280902753920*x^9 - 2350198326509186389*x^8 - 4124312348255593
984*x^7 + 1013303893827450652*x^6 + 798797730254046208*x^5 - 145619826687334470*x^4 - 83828069684567680*x^3 +
1431075537830988*x^2 + 2060726880350592*x + 33216827121477)) - 13^(1/4)*(3*sqrt(13)*sqrt(2)*x - 13*sqrt(2)*x)*
sqrt(3*sqrt(13) + 13)*log(1300*(52*x^4 + 52*x^3 + 13^(1/4)*sqrt(x^4 + x^3 - x^2 - x + 1)*(sqrt(13)*sqrt(2)*(x^
2 + 2*x - 1) + sqrt(2)*(5*x^2 - 2*x - 5))*sqrt(3*sqrt(13) + 13) - 52*x^2 + sqrt(13)*(17*x^4 + 20*x^3 - 26*x^2
- 20*x + 17) - 52*x + 52)/(x^4 + 2*x^2 + 1)) + 13^(1/4)*(3*sqrt(13)*sqrt(2)*x - 13*sqrt(2)*x)*sqrt(3*sqrt(13)
+ 13)*log(1300*(52*x^4 + 52*x^3 - 13^(1/4)*sqrt(x^4 + x^3 - x^2 - x + 1)*(sqrt(13)*sqrt(2)*(x^2 + 2*x - 1) + s
qrt(2)*(5*x^2 - 2*x - 5))*sqrt(3*sqrt(13) + 13) - 52*x^2 + sqrt(13)*(17*x^4 + 20*x^3 - 26*x^2 - 20*x + 17) - 5
2*x + 52)/(x^4 + 2*x^2 + 1)) - 104*x*log(-(2*x^2 + x + 2*sqrt(x^4 + x^3 - x^2 - x + 1) - 2)/x) - 208*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*} \int \frac {\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)*(x^4+x^3-x^2-x+1)^(1/2)/x^2/(x^2+1),x, algorithm="giac")
[Out]
integrate(sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 - 1)/((x^2 + 1)*x^2), x)
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maple [B] time = 2.57, size = 648, normalized size = 2.55
| | |
| method |
result |
size |
| | |
| trager |
\(\frac {\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) \ln \left (-\frac {-51 \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{5} x^{2}+51 \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{5}-362 x^{2} \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}-136 x \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}+448 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+362 \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}-247 x^{2} \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )-104 x \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )+1872 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+247 \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )}{\left (x \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+5 x -1\right )^{2}}\right )}{2}-\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \ln \left (\frac {51 \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{4} x^{2}-51 \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{4} \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )+250 \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} x^{2}-136 \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} x +448 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-250 \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )-89 \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) x^{2}-712 \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) x +816 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+89 \RootOf \left (\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )}{\left (x \RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-\RootOf \left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+x -5\right )^{2}}\right )}{2}+\frac {\ln \left (-\frac {2 x^{2}+2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+x -2}{x}\right )}{2}\) |
\(648\) |
| risch |
\(\text {Expression too large to display}\) |
\(186639\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(286046\) |
| default |
\(\text {Expression too large to display}\) |
\(339777\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-1)*(x^4+x^3-x^2-x+1)^(1/2)/x^2/(x^2+1),x,method=_RETURNVERBOSE)
[Out]
(x^4+x^3-x^2-x+1)^(1/2)/x-1/2*RootOf(_Z^4+6*_Z^2+13)*ln(-(-51*RootOf(_Z^4+6*_Z^2+13)^5*x^2+51*RootOf(_Z^4+6*_Z
^2+13)^5-362*x^2*RootOf(_Z^4+6*_Z^2+13)^3-136*x*RootOf(_Z^4+6*_Z^2+13)^3+448*(x^4+x^3-x^2-x+1)^(1/2)*RootOf(_Z
^4+6*_Z^2+13)^2+362*RootOf(_Z^4+6*_Z^2+13)^3-247*x^2*RootOf(_Z^4+6*_Z^2+13)-104*x*RootOf(_Z^4+6*_Z^2+13)+1872*
(x^4+x^3-x^2-x+1)^(1/2)+247*RootOf(_Z^4+6*_Z^2+13))/(x*RootOf(_Z^4+6*_Z^2+13)^2-RootOf(_Z^4+6*_Z^2+13)^2+5*x-1
)^2)-1/2*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6)*ln((51*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6)*RootOf(_Z^4+6*
_Z^2+13)^4*x^2-51*RootOf(_Z^4+6*_Z^2+13)^4*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6)+250*RootOf(RootOf(_Z^4+6*_Z
^2+13)^2+_Z^2+6)*RootOf(_Z^4+6*_Z^2+13)^2*x^2-136*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6)*RootOf(_Z^4+6*_Z^2+1
3)^2*x+448*(x^4+x^3-x^2-x+1)^(1/2)*RootOf(_Z^4+6*_Z^2+13)^2-250*RootOf(_Z^4+6*_Z^2+13)^2*RootOf(RootOf(_Z^4+6*
_Z^2+13)^2+_Z^2+6)-89*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6)*x^2-712*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6)*
x+816*(x^4+x^3-x^2-x+1)^(1/2)+89*RootOf(RootOf(_Z^4+6*_Z^2+13)^2+_Z^2+6))/(x*RootOf(_Z^4+6*_Z^2+13)^2-RootOf(_
Z^4+6*_Z^2+13)^2+x-5)^2)+1/2*ln(-(2*x^2+2*(x^4+x^3-x^2-x+1)^(1/2)+x-2)/x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)*(x^4+x^3-x^2-x+1)^(1/2)/x^2/(x^2+1),x, algorithm="maxima")
[Out]
integrate(sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 - 1)/((x^2 + 1)*x^2), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+x^3-x^2-x+1}}{x^2\,\left (x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^2 - 1)*(x^3 - x^2 - x + x^4 + 1)^(1/2))/(x^2*(x^2 + 1)),x)
[Out]
int(((x^2 - 1)*(x^3 - x^2 - x + x^4 + 1)^(1/2))/(x^2*(x^2 + 1)), 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 ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}{x^{2} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2-1)*(x**4+x**3-x**2-x+1)**(1/2)/x**2/(x**2+1),x)
[Out]
Integral((x - 1)*(x + 1)*sqrt(x**4 + x**3 - x**2 - x + 1)/(x**2*(x**2 + 1)), x)
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