3.4.74 \(\int \frac {-2-2 x+x^2}{(-1+3 x+x^2) \sqrt {-1+x^3}} \, dx\) [374]
Optimal. Leaf size=31 \[ -\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]
[Out]
-2^(1/2)*arctan(2^(1/2)*(x^3-1)^(1/2)/(x^2+x+1))
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Rubi [A]
time = 0.05, antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2170, 209}
\begin {gather*} \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} (1-x)}{\sqrt {x^3-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-2 - 2*x + x^2)/((-1 + 3*x + x^2)*Sqrt[-1 + x^3]),x]
[Out]
Sqrt[2]*ArcTan[(Sqrt[2]*(1 - x))/Sqrt[-1 + x^3]]
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 2170
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /;
FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h,
0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
Rubi steps
\begin {align*} \int \frac {-2-2 x+x^2}{\left (-1+3 x+x^2\right ) \sqrt {-1+x^3}} \, dx &=4 \text {Subst}\left (\int \frac {1}{2+4 x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right )\\ &=\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-1+x^3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 31, normalized size = 1.00 \begin {gather*} -\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-1+x^3}}{1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-2 - 2*x + x^2)/((-1 + 3*x + x^2)*Sqrt[-1 + x^3]),x]
[Out]
-(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[-1 + x^3])/(1 + x + x^2)])
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.97, size = 1641, normalized size = 52.94
| | |
| method |
result |
size |
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| trager |
\(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{3}-1}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{2}+3 x -1}\right )}{2}\) |
\(58\) |
| default |
\(\text {Expression too large to display}\) |
\(1641\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(1850\) |
| | |
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-2*x-2)/(x^2+3*x-1)/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
[Out]
2*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*(
(x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(
(3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-3/2*13^(1/2)*(-1/(-3/2-1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/2))*x
)^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*
I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))
*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3
/2-1/2*I*3^(1/2)))^(1/2))-1/2*I*13^(1/2)*(-1/(-3/2-1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(3/2-1/2*
I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/
2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*EllipticPi(((-1+x)/
(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(
1/2))*3^(1/2)+15/2*(-1/(-3/2-1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-
1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I
/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^
(1/2),(3/2+1/2*I*3^(1/2))/(5/2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+5/2*I*(-1/(-3/2-
1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2
*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))
^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/
2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)+3/2*13^(1/2)*(-1/(-3/2-1/2*I*3^(1/2))
+1/(-3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^
(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)
^(1/2)/(5/2+1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)
),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+1/2*I*13^(1/2)*(-1/(-3/2-1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/
2))*x)^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2
+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2+1/2*13^(
1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3/2+1/2*I*3^(1/2
))/(3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)+15/2*(-1/(-3/2-1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(3/2-1
/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/
(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2+1/2*13^(1/2))*EllipticPi(((-1+
x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2))
)^(1/2))+5/2*I*(-1/(-3/2-1/2*I*3^(1/2))+1/(-3/2-1/2*I*3^(1/2))*x)^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*
I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/
2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2+1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2
),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)
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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^2-2*x-2)/(x^2+3*x-1)/(x^3-1)^(1/2),x, algorithm="maxima")
[Out]
integrate((x^2 - 2*x - 2)/(sqrt(x^3 - 1)*(x^2 + 3*x - 1)), x)
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Fricas [A]
time = 0.37, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - x + 3\right )}}{4 \, \sqrt {x^{3} - 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-2*x-2)/(x^2+3*x-1)/(x^3-1)^(1/2),x, algorithm="fricas")
[Out]
1/2*sqrt(2)*arctan(1/4*sqrt(2)*(x^2 - x + 3)/sqrt(x^3 - 1))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 2 x - 2}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 3 x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2-2*x-2)/(x**2+3*x-1)/(x**3-1)**(1/2),x)
[Out]
Integral((x**2 - 2*x - 2)/(sqrt((x - 1)*(x**2 + x + 1))*(x**2 + 3*x - 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^2-2*x-2)/(x^2+3*x-1)/(x^3-1)^(1/2),x, algorithm="giac")
[Out]
integrate((x^2 - 2*x - 2)/(sqrt(x^3 - 1)*(x^2 + 3*x - 1)), x)
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Mupad [B]
time = 0.10, size = 273, normalized size = 8.81 \begin {gather*} \frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {\sqrt {13}}{2}+\frac {5}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {\sqrt {13}}{2}-\frac {5}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(2*x - x^2 + 2)/((x^3 - 1)^(1/2)*(3*x + x^2 - 1)),x)
[Out]
((3^(1/2)*1i + 3)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(
1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(13^(1/2)/
2 + 5/2), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) - ell
ipticF(asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) + ellipt
icPi(-((3^(1/2)*1i)/2 + 3/2)/(13^(1/2)/2 - 5/2), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)
/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)
*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)
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