3.28.39 \(\int \frac {(-1+x^2) \sqrt {1-x-x^2+x^3+x^4}}{x^2 (1+x^2)} \, dx\) [2739]

Optimal. Leaf size=254 \[ \frac {\sqrt {1-x-x^2+x^3+x^4}}{x}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )+\text {RootSum}\left [9+12 \text {$\#$1}+62 \text {$\#$1}^2-4 \text {$\#$1}^3+\text {$\#$1}^4\& ,\frac {9 \log (x)-9 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right )-10 \log (x) \text {$\#$1}+10 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right ) \text {$\#$1}^2}{3+31 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [F]
time = 0.47, 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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.21, size = 229, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (\frac {2 \sqrt {1-x-x^2+x^3+x^4}}{x}+\log (x)-\log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )-\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 ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 3.24, size = 339777, normalized size = 1337.70

method	result	size
trager	\(\text {Expression too large to display}\)	\(651\)
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]

result too large to display

________________________________________________________________________________________

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^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)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 2.62, size = 4897, normalized size = 19.28 \begin {gather*} \text {Too large to display} \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="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...

________________________________________________________________________________________

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)

________________________________________________________________________________________

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^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)

________________________________________________________________________________________

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)

________________________________________________________________________________________