3.10.85 \(\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} (-1-2 x+x^2+3 x^3)} \, dx\)
Optimal. Leaf size=75 \[ \frac {\left ((x-1)^3\right )^{3/4} \text {RootSum}\left [3 \text {$\#$1}^{12}+10 \text {$\#$1}^8+9 \text {$\#$1}^4+1\& ,\frac {\log \left (\sqrt [4]{x-1}-\text {$\#$1}\right )}{9 \text {$\#$1}^{11}+20 \text {$\#$1}^7+9 \text {$\#$1}^3}\& \right ]}{(x-1)^{9/4}} \]
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Rubi [F] time = 0.12, 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}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)),x]
[Out]
Defer[Int][1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)), x]
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx &=\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx\\ \end {align*}
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Mathematica [A] time = 0.08, size = 75, normalized size = 1.00 \begin {gather*} \frac {(x-1)^{3/4} \text {RootSum}\left [3 \text {$\#$1}^{12}+10 \text {$\#$1}^8+9 \text {$\#$1}^4+1\&,\frac {\log \left (\sqrt [4]{x-1}-\text {$\#$1}\right )}{9 \text {$\#$1}^{11}+20 \text {$\#$1}^7+9 \text {$\#$1}^3}\&\right ]}{\sqrt [4]{(x-1)^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)),x]
[Out]
((-1 + x)^(3/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , Log[(-1 + x)^(1/4) - #1]/(9*#1^3 + 20*#1^7 + 9*#1^1
1) & ])/((-1 + x)^3)^(1/4)
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IntegrateAlgebraic [A] time = 10.43, size = 75, normalized size = 1.00 \begin {gather*} \frac {\left ((-1+x)^3\right )^{3/4} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\&\right ]}{(-1+x)^{9/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)),x]
[Out]
(((-1 + x)^3)^(3/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , Log[(-1 + x)^(1/4) - #1]/(9*#1^3 + 20*#1^7 + 9*
#1^11) & ])/(-1 + x)^(9/4)
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fricas [B] time = 1.40, size = 6190, normalized size = 82.53
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x, algorithm="fricas")
[Out]
2/31*sqrt(31)*sqrt(2)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I
*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210
333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 +
390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) +
1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)*arctan(1/31037243945331168*sqrt(2)*(14386551444
*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 - sqrt(1864509)*sqrt(1/31)*(643*sqr
t(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(6
21503*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 389166*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(
1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) +
62*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3
) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*
sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(643*((1
/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqr
t(93) - 6210333)^(1/3) + 260)*(x - 1) - 112374*x + 112374) + 46529388*sqrt(31)*(x - 1))*sqrt(((3239*(x^2 - 2*x
+ 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(6
21503*sqrt(93) - 6210333)^(1/3) + 260)^2 - 551100*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1
/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 159338016
*x^2 + 2*(3239*sqrt(31)*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 1975320*sqrt(31)*(x^2 - 2*x + 1))*s
qrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) +
1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt
(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) + 318676032*
x - 159338016)*sqrt((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sq
rt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333
)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390
/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)
/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520) + 8323168176*sqrt(x^3 - 3*x^2 + 3*x - 1))/(x^2 - 2*x + 1
)) - 8707242113928*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*
sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) + 1387194696*sqrt(-
3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(6
21503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) +
1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(643*(x^3 - 3*x^2
+ 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqr
t(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 112374*(x^3 - 3*x^2 + 3*x - 1)^(1/4)) + 1041053552285904*
sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4))*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(6215
03*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)
^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^
(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)/(x - 1)) - 2/31*sqrt(31)*sqrt
(2)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621
503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqr
t(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3
)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(9
3) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)*arctan(1/31037243945331168*(sqrt(1864509)*sqrt(2)*sqrt(1/31)*(643
*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1
)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 389166*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 621033
3)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1
) - 62*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sq
rt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*
(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(643
*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503
*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) - 112374*x + 112374) + 46529388*sqrt(31)*(x - 1))*((1/18)^(1/3)*(621
503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333
)^(1/3) + 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/1
8)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6
210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) +
74864/31) - 520)^(1/4)*sqrt(((3239*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3)
+ 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 - 551100*(x^2 - 2*x + 1)
*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503
*sqrt(93) - 6210333)^(1/3) + 260) - 159338016*x^2 - 2*(3239*sqrt(31)*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqr
t(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3)
+ 260) - 1975320*sqrt(31)*(x^2 - 2*x + 1))*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sq
rt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/
3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(
93) - 6210333)^(1/3) + 74864/31) + 318676032*x - 159338016)*sqrt((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3
)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)*sqrt(-3
/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(62
1503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) +
1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520) + 832316817
6*sqrt(x^3 - 3*x^2 + 3*x - 1))/(x^2 - 2*x + 1)) - 22374108*sqrt(2)*(643*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)
*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503
*sqrt(93) - 6210333)^(1/3) + 260)^2 - 389166*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt
(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3)
+ 260) - 62*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*
(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(
1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)
*(643*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(
1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 112374*(x^3 - 3*x^2 + 3*x - 1)^(1/4)) +
46529388*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4))*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3)
+ 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)*sqrt(-3/124*((1/18)^(
1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93)
- 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/
31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4))/(x - 1)) - 1/62*sq
rt(31)*sqrt(2)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3
) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1
/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*
(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(62
1503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)*log(1/3844*(sqrt(2)*(54235*sqrt(31)*((1/18)^(1/3)*(621
503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333
)^(1/3) + 260)^2*(x - 1) - 40706304*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1)
+ 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) - 62*sqrt(-3/124*((1/18)
^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93
) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 1413594
0/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(54235*((1/18)^(1/3)*(621503*s
qrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/
3) + 260)*(x - 1) - 1596996*x + 1596996) + 9774338688*sqrt(31)*(x - 1))*((1/18)^(1/3)*(621503*sqrt(93) - 62103
33)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)
*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3)
+ 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sq
rt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(3/
4) + 4644327842208*(x^3 - 3*x^2 + 3*x - 1)^(1/4))/(x - 1)) + 1/62*sqrt(31)*sqrt(2)*((1/18)^(1/3)*(621503*sqrt(
93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) +
2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*
(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(
1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)
- 520)^(1/4)*log(-1/3844*(sqrt(2)*(54235*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3)
+ 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 40706304*sqrt(
31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621
503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) - 62*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*
(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/1
8)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503
*sqrt(93) - 6210333)^(1/3) + 74864/31)*(54235*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1)
+ 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) - 1596996*x + 1596996)
+ 9774338688*sqrt(31)*(x - 1))*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)
^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(9
3) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) +
260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*
sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(3/4) - 4644327842208*(x^3 - 3*x^2 + 3*x - 1
)^(1/4))/(x - 1)) - 1/62*sqrt(31)*sqrt(2)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 3
6246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(62
1503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 621033
3)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18
)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)*log(1/3844*(sqrt(2)*(54235*
sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)
/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 40706304*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 62103
33)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x -
1) + 62*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*s
qrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)
*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(54
235*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621
503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) - 1596996*x + 1596996) + 9774338688*sqrt(31)*(x - 1))*((1/18)^(1/
3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) -
6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(
93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(
1/3) + 74864/31) - 520)^(3/4) + 4644327842208*(x^3 - 3*x^2 + 3*x - 1)^(1/4))/(x - 1)) + 1/62*sqrt(31)*sqrt(2)*
((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*
sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3)
+ 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(6
21503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) -
6210333)^(1/3) + 74864/31) - 520)^(1/4)*log(-1/3844*(sqrt(2)*(54235*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) -
6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)
^2*(x - 1) - 40706304*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)
^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) + 62*sqrt(-3/124*((1/18)^(1/3)*(621503
*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(
1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2
/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(54235*((1/18)^(1/3)*(621503*sqrt(93) - 6210
333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x -
1) - 1596996*x + 1596996) + 9774338688*sqrt(31)*(x - 1))*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*
sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*(
(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*s
qrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 1
4135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(3/4) - 464432784
2208*(x^3 - 3*x^2 + 3*x - 1)^(1/4))/(x - 1)) + 4*(-1/1922*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*s
qrt(3) + 1) - 18123/961*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 130/961)^(1/4)*arctan
(-1/125150177198916*(7193275722*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*
(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 - sqrt(186450
9)*(643*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/
(621503*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 389166*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-
I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) + 4777239
4*x - 47772394)*sqrt(-((3239*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1)
+ 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 - 551100*(x^2 - 2*x + 1)*((1/1
8)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(
93) - 6210333)^(1/3) + 260) - 112103788*x^2 + 224207576*x - 112103788)*sqrt(-1/1922*(1/18)^(1/3)*(621503*sqrt(
93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) - 18123/961*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/
3) - 130/961) - 67122324*sqrt(x^3 - 3*x^2 + 3*x - 1))/(x^2 - 2*x + 1)) - 4353621056964*(x^3 - 3*x^2 + 3*x - 1)
^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(
621503*sqrt(93) - 6210333)^(1/3) + 260) + 534432351387276*(x^3 - 3*x^2 + 3*x - 1)^(1/4))*(-1/1922*(1/18)^(1/3)
*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) - 18123/961*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93)
- 6210333)^(1/3) - 130/961)^(1/4)/(x - 1)) + (-1/1922*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(
3) + 1) - 18123/961*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 130/961)^(1/4)*log(((5423
5*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(62150
3*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 40706304*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sq
rt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) + 4178325676*
x - 4178325676)*(-1/1922*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) - 18123/961*(1/18)^(2
/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 130/961)^(3/4) + 604100916*(x^3 - 3*x^2 + 3*x - 1)^(1/
4))/(x - 1)) - (-1/1922*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) - 18123/961*(1/18)^(2/
3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 130/961)^(1/4)*log(-((54235*((1/18)^(1/3)*(621503*sqrt(
93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) +
260)^2*(x - 1) - 40706304*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/
3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) + 4178325676*x - 4178325676)*(-1/1922*(1/1
8)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) - 18123/961*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*s
qrt(93) - 6210333)^(1/3) - 130/961)^(3/4) - 604100916*(x^3 - 3*x^2 + 3*x - 1)^(1/4))/(x - 1))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x, algorithm="giac")
[Out]
integrate(1/((3*x^3 + x^2 - 2*x - 1)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)), x)
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}} \left (3 x^{3}+x^{2}-2 x -1\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x)
[Out]
int(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x^{3} + x^{2} - 2 \, x - 1\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x, algorithm="maxima")
[Out]
integrate(1/((3*x^3 + x^2 - 2*x - 1)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{{\left (x^3-3\,x^2+3\,x-1\right )}^{1/4}\,\left (-3\,x^3-x^2+2\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-1/((3*x - 3*x^2 + x^3 - 1)^(1/4)*(2*x - x^2 - 3*x^3 + 1)),x)
[Out]
-int(1/((3*x - 3*x^2 + x^3 - 1)^(1/4)*(2*x - x^2 - 3*x^3 + 1)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x^{3} + x^{2} - 2 x - 1\right ) \sqrt [4]{\left (x - 1\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x**3-3*x**2+3*x-1)**(1/4)/(3*x**3+x**2-2*x-1),x)
[Out]
Integral(1/((3*x**3 + x**2 - 2*x - 1)*((x - 1)**3)**(1/4)), x)
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