3.10.65 \(\int \frac {\sqrt {-1+x^2+x^4+x^6} (1+x^4+2 x^6)}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx\) [965]

Optimal. Leaf size=73 \[ -\frac {1}{2} \sqrt {1+i} \text {ArcTan}\left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^4+x^6}}\right )-\frac {1}{2} \sqrt {1-i} \text {ArcTan}\left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^4+x^6}}\right ) \]

[Out]

-1/2*(1+I)^(1/2)*arctan((-1-I)^(1/2)*x/(x^6+x^4+x^2-1)^(1/2))-1/2*(1-I)^(1/2)*arctan((-1+I)^(1/2)*x/(x^6+x^4+x
^2-1)^(1/2))

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Rubi [F]
time = 0.63, 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 {\sqrt {-1+x^2+x^4+x^6} \left (1+x^4+2 x^6\right )}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[-1 + x^2 + x^4 + x^6]*(1 + x^4 + 2*x^6))/(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12),x]

[Out]

Defer[Int][Sqrt[-1 + x^2 + x^4 + x^6]/(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12), x] + Defer[Int][(x^4*Sqrt[-1 +
x^2 + x^4 + x^6])/(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12), x] + 2*Defer[Int][(x^6*Sqrt[-1 + x^2 + x^4 + x^6])/
(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^2+x^4+x^6} \left (1+x^4+2 x^6\right )}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx &=\int \left (\frac {\sqrt {-1+x^2+x^4+x^6}}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}}+\frac {x^4 \sqrt {-1+x^2+x^4+x^6}}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}}+\frac {2 x^6 \sqrt {-1+x^2+x^4+x^6}}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}}\right ) \, dx\\ &=2 \int \frac {x^6 \sqrt {-1+x^2+x^4+x^6}}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx+\int \frac {\sqrt {-1+x^2+x^4+x^6}}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx+\int \frac {x^4 \sqrt {-1+x^2+x^4+x^6}}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 73, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {1+i} \text {ArcTan}\left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^4+x^6}}\right )-\frac {1}{2} \sqrt {1-i} \text {ArcTan}\left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^4+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + x^2 + x^4 + x^6]*(1 + x^4 + 2*x^6))/(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12),x]

[Out]

-1/2*(Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/Sqrt[-1 + x^2 + x^4 + x^6]]) - (Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/
Sqrt[-1 + x^2 + x^4 + x^6]])/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 5.56, size = 589, normalized size = 8.07

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2} x^{6}+64 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{4} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2} x^{4}+12 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2} x^{2}+16 \sqrt {x^{6}+x^{4}+x^{2}-1}\, \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2} x -4 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+2\right )+\sqrt {x^{6}+x^{4}+x^{2}-1}\, x}{-x^{6}+16 x^{2} \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}-x^{4}+x^{2}+1}\right )}{4}-\RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{3} x^{6}-256 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{5} x^{2}+16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{3} x^{4}+2 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right ) x^{6}-16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{3} x^{2}+2 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right ) x^{4}+16 \sqrt {x^{6}+x^{4}+x^{2}-1}\, \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2} x -16 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{3}+2 x^{2} \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )+\sqrt {x^{6}+x^{4}+x^{2}-1}\, x -2 \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )}{x^{6}+16 x^{2} \RootOf \left (128 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+x^{2}-1}\right )\)	\(589\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+x^4+x^2-1)^(1/2)*(2*x^6+x^4+1)/(x^12+2*x^10+x^8-2*x^6-x^4+1),x,method=_RETURNVERBOSE)

[Out]

-1/4*RootOf(_Z^2+16*RootOf(128*_Z^4+16*_Z^2+1)^2+2)*ln((4*RootOf(_Z^2+16*RootOf(128*_Z^4+16*_Z^2+1)^2+2)*RootO
f(128*_Z^4+16*_Z^2+1)^2*x^6+64*RootOf(_Z^2+16*RootOf(128*_Z^4+16*_Z^2+1)^2+2)*RootOf(128*_Z^4+16*_Z^2+1)^4*x^2
+4*RootOf(_Z^2+16*RootOf(128*_Z^4+16*_Z^2+1)^2+2)*RootOf(128*_Z^4+16*_Z^2+1)^2*x^4+12*RootOf(_Z^2+16*RootOf(12
8*_Z^4+16*_Z^2+1)^2+2)*RootOf(128*_Z^4+16*_Z^2+1)^2*x^2+16*(x^6+x^4+x^2-1)^(1/2)*RootOf(128*_Z^4+16*_Z^2+1)^2*
x-4*RootOf(128*_Z^4+16*_Z^2+1)^2*RootOf(_Z^2+16*RootOf(128*_Z^4+16*_Z^2+1)^2+2)+(x^6+x^4+x^2-1)^(1/2)*x)/(-x^6
+16*x^2*RootOf(128*_Z^4+16*_Z^2+1)^2-x^4+x^2+1))-RootOf(128*_Z^4+16*_Z^2+1)*ln(-(16*RootOf(128*_Z^4+16*_Z^2+1)
^3*x^6-256*RootOf(128*_Z^4+16*_Z^2+1)^5*x^2+16*RootOf(128*_Z^4+16*_Z^2+1)^3*x^4+2*RootOf(128*_Z^4+16*_Z^2+1)*x
^6-16*RootOf(128*_Z^4+16*_Z^2+1)^3*x^2+2*RootOf(128*_Z^4+16*_Z^2+1)*x^4+16*(x^6+x^4+x^2-1)^(1/2)*RootOf(128*_Z
^4+16*_Z^2+1)^2*x-16*RootOf(128*_Z^4+16*_Z^2+1)^3+2*x^2*RootOf(128*_Z^4+16*_Z^2+1)+(x^6+x^4+x^2-1)^(1/2)*x-2*R
ootOf(128*_Z^4+16*_Z^2+1))/(x^6+16*x^2*RootOf(128*_Z^4+16*_Z^2+1)^2+x^4+x^2-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+x^4+x^2-1)^(1/2)*(2*x^6+x^4+1)/(x^12+2*x^10+x^8-2*x^6-x^4+1),x, algorithm="maxima")

[Out]

integrate((2*x^6 + x^4 + 1)*sqrt(x^6 + x^4 + x^2 - 1)/(x^12 + 2*x^10 + x^8 - 2*x^6 - x^4 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6596 vs. \(2 (49) = 98\).
time = 3.42, size = 6596, normalized size = 90.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+x^4+x^2-1)^(1/2)*(2*x^6+x^4+1)/(x^12+2*x^10+x^8-2*x^6-x^4+1),x, algorithm="fricas")

[Out]

1/32*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2)*log(2*(8*x^8 + 8*x^6 + 8*x^4 + 2*2^(1/4)*(x^7 + x^5 + sqrt(2)*x
^3 + x^3 - x)*sqrt(x^6 + x^4 + x^2 - 1)*sqrt(2*sqrt(2) + 4) - 8*x^2 + sqrt(2)*(x^12 + 2*x^10 + 5*x^8 + 2*x^6 +
3*x^4 - 4*x^2 + 1))/(x^12 + 2*x^10 + x^8 - 2*x^6 - x^4 + 1)) - 1/32*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2)
*log(2*(8*x^8 + 8*x^6 + 8*x^4 - 2*2^(1/4)*(x^7 + x^5 + sqrt(2)*x^3 + x^3 - x)*sqrt(x^6 + x^4 + x^2 - 1)*sqrt(2
*sqrt(2) + 4) - 8*x^2 + sqrt(2)*(x^12 + 2*x^10 + 5*x^8 + 2*x^6 + 3*x^4 - 4*x^2 + 1))/(x^12 + 2*x^10 + x^8 - 2*
x^6 - x^4 + 1)) + 1/8*2^(3/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(4*x^72 + 48*x^70 + 296*x^68 + 1184*x^66 + 4100*x
^64 + 14336*x^62 + 48576*x^60 + 136416*x^58 + 243584*x^56 + 55200*x^54 - 1076416*x^52 - 3184640*x^50 - 3948592
*x^48 + 641728*x^46 + 10292800*x^44 + 15128864*x^42 + 3787784*x^40 - 17971776*x^38 - 25671216*x^36 - 5595392*x
^34 + 21212056*x^32 + 22744448*x^30 - 89792*x^28 - 16094688*x^26 - 9256416*x^24 + 4023008*x^22 + 6035840*x^20
+ 544896*x^18 - 1868656*x^16 - 554560*x^14 + 332224*x^12 + 116192*x^10 - 36524*x^8 - 5232*x^6 + 840*x^4 - 32*x
^2 + 4*sqrt(x^6 + x^4 + x^2 - 1)*(2^(3/4)*(4*x^67 + 44*x^65 + 254*x^63 + 956*x^61 + 2382*x^59 + 3356*x^57 + 35
0*x^55 - 7792*x^53 - 5766*x^51 + 40296*x^49 + 137174*x^47 + 186756*x^45 + 20238*x^43 - 366284*x^41 - 611162*x^
39 - 279864*x^37 + 486370*x^35 + 879240*x^33 + 368218*x^31 - 485916*x^29 - 681958*x^27 - 123804*x^25 + 351834*
x^23 + 245984*x^21 - 61090*x^19 - 118680*x^17 - 9422*x^15 + 30876*x^13 + 4842*x^11 - 5236*x^9 + 194*x^7 + 72*x
^5 + 34*x^3 - sqrt(2)*(3*x^67 + 33*x^65 + 188*x^63 + 692*x^61 + 1414*x^59 - 98*x^57 - 11242*x^55 - 39830*x^53
- 72700*x^51 - 52668*x^49 + 87784*x^47 + 304536*x^45 + 361362*x^43 + 19402*x^41 - 562182*x^39 - 769994*x^37 -
196922*x^35 + 665266*x^33 + 842764*x^31 + 131812*x^29 - 569670*x^27 - 479358*x^25 + 68762*x^23 + 294150*x^21 +
92604*x^19 - 86324*x^17 - 54576*x^15 + 14160*x^13 + 14126*x^11 - 2090*x^9 - 1674*x^7 + 314*x^5 + 23*x^3 - 3*x
) - 4*x) + 32*2^(1/4)*(7*x^63 + 70*x^61 + 419*x^59 + 1706*x^57 + 4942*x^55 + 9908*x^53 + 11825*x^51 + 580*x^49
- 28977*x^47 - 58322*x^45 - 45399*x^43 + 29706*x^41 + 111540*x^39 + 100724*x^37 - 20967*x^35 - 131220*x^33 -
100923*x^31 + 30198*x^29 + 100465*x^27 + 45314*x^25 - 34938*x^23 - 41936*x^21 - 1689*x^19 + 16480*x^17 + 5389*
x^15 - 3690*x^13 - 1861*x^11 + 586*x^9 + 288*x^7 - 104*x^5 + 7*x^3 - sqrt(2)*(5*x^63 + 50*x^61 + 298*x^59 + 12
07*x^57 + 3407*x^55 + 6367*x^53 + 6040*x^51 - 4429*x^49 - 25315*x^47 - 38326*x^45 - 15109*x^43 + 43760*x^41 +
83353*x^39 + 43063*x^37 - 53901*x^35 - 101512*x^33 - 39859*x^31 + 55764*x^29 + 71846*x^27 + 8493*x^25 - 38913*
x^23 - 24125*x^21 + 7816*x^19 + 12629*x^17 + 1009*x^15 - 3464*x^13 - 795*x^11 + 596*x^9 + 129*x^7 - 73*x^5 + 5
*x^3)))*sqrt(2*sqrt(2) + 4) - sqrt(2)*(8*(96*x^63 + 960*x^61 + 5440*x^59 + 20640*x^57 + 49056*x^55 + 51360*x^5
3 - 86656*x^51 - 440800*x^49 - 748192*x^47 - 344896*x^45 + 1050528*x^43 + 2298624*x^41 + 1470048*x^39 - 148080
0*x^37 - 3533152*x^35 - 1909504*x^33 + 1775712*x^31 + 3135616*x^29 + 903360*x^27 - 1581856*x^25 - 1461344*x^23
+ 115232*x^21 + 733312*x^19 + 220384*x^17 - 187936*x^15 - 105984*x^13 + 26720*x^11 + 22144*x^9 - 3744*x^7 - 1
120*x^5 + 96*x^3 + sqrt(2)*(2*x^67 + 22*x^65 + 118*x^63 + 388*x^61 + 242*x^59 - 4208*x^57 - 24094*x^55 - 72564
*x^53 - 137454*x^51 - 143968*x^49 + 17718*x^47 + 348676*x^45 + 593306*x^43 + 379088*x^41 - 322846*x^39 - 92123
6*x^37 - 716514*x^35 + 200860*x^33 + 869330*x^31 + 583276*x^29 - 209930*x^27 - 538448*x^25 - 193674*x^23 + 190
852*x^21 + 173798*x^19 - 18704*x^17 - 68382*x^15 - 6708*x^13 + 16990*x^11 + 2096*x^9 - 3418*x^7 + 580*x^5 + 8*
x^3 - sqrt(2)*(x^67 + 11*x^65 + 63*x^63 + 234*x^61 + 817*x^59 + 2980*x^57 + 10977*x^55 + 33730*x^53 + 77885*x^
51 + 120232*x^49 + 83383*x^47 - 117086*x^45 - 415651*x^43 - 508756*x^41 - 102871*x^39 + 599162*x^37 + 876907*x
^35 + 288542*x^33 - 600475*x^31 - 787586*x^29 - 136717*x^27 + 458364*x^25 + 357051*x^23 - 66490*x^21 - 195937*
x^19 - 40496*x^17 + 57109*x^15 + 21718*x^13 - 11153*x^11 - 4332*x^9 + 2219*x^7 - 226*x^5 + 8*x^3 - x) - 2*x) -
64*sqrt(2)*(x^63 + 10*x^61 + 58*x^59 + 227*x^57 + 647*x^55 + 1339*x^53 + 1888*x^51 + 1247*x^49 - 1539*x^47 -
5654*x^45 - 7517*x^43 - 3320*x^41 + 5833*x^39 + 12123*x^37 + 8079*x^35 - 3712*x^33 - 11255*x^31 - 6900*x^29 +
3078*x^27 + 6825*x^25 + 2263*x^23 - 2521*x^21 - 2152*x^19 + 321*x^17 + 873*x^15 + 40*x^13 - 227*x^11 - 12*x^9
+ 49*x^7 - 13*x^5 + x^3))*sqrt(x^6 + x^4 + x^2 - 1) + (2^(3/4)*(2*x^72 + 24*x^70 + 136*x^68 + 460*x^66 + 346*x
^64 - 5120*x^62 - 32464*x^60 - 110424*x^58 - 252208*x^56 - 389560*x^54 - 341144*x^52 + 79120*x^50 + 799728*x^4
8 + 1350768*x^46 + 1111936*x^44 - 74504*x^42 - 1522780*x^40 - 2032504*x^38 - 965272*x^36 + 913656*x^34 + 18745
48*x^32 + 1026656*x^30 - 569936*x^28 - 1124264*x^26 - 356512*x^24 + 426616*x^22 + 348824*x^20 - 67920*x^18 - 1
44192*x^16 + 6448*x^14 + 39840*x^12 - 3832*x^10 - 9558*x^8 + 4384*x^6 - 624*x^4 - 4*x^2 - sqrt(2)*(x^72 + 12*x
^70 + 70*x^68 + 252*x^66 + 797*x^64 + 2888*x^62...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 1\right ) \left (2 x^{4} - x^{2} + 1\right ) \sqrt {x^{6} + x^{4} + x^{2} - 1}}{x^{12} + 2 x^{10} + x^{8} - 2 x^{6} - x^{4} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+x**4+x**2-1)**(1/2)*(2*x**6+x**4+1)/(x**12+2*x**10+x**8-2*x**6-x**4+1),x)

[Out]

Integral((x**2 + 1)*(2*x**4 - x**2 + 1)*sqrt(x**6 + x**4 + x**2 - 1)/(x**12 + 2*x**10 + x**8 - 2*x**6 - x**4 +
1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+x^4+x^2-1)^(1/2)*(2*x^6+x^4+1)/(x^12+2*x^10+x^8-2*x^6-x^4+1),x, algorithm="giac")

[Out]

integrate((2*x^6 + x^4 + 1)*sqrt(x^6 + x^4 + x^2 - 1)/(x^12 + 2*x^10 + x^8 - 2*x^6 - x^4 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^6+x^4+1\right )\,\sqrt {x^6+x^4+x^2-1}}{x^{12}+2\,x^{10}+x^8-2\,x^6-x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 2*x^6 + 1)*(x^2 + x^4 + x^6 - 1)^(1/2))/(x^8 - 2*x^6 - x^4 + 2*x^10 + x^12 + 1),x)

[Out]

int(((x^4 + 2*x^6 + 1)*(x^2 + x^4 + x^6 - 1)^(1/2))/(x^8 - 2*x^6 - x^4 + 2*x^10 + x^12 + 1), x)

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