3.8.23 \(\int \frac {\sqrt {-1+x^2-2 x^3+x^4} (1-x^3+x^4)}{(-1-2 x^3+x^4) (-2-x^2-4 x^3+2 x^4)} \, dx\)
Optimal. Leaf size=57 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-2 x^3+x^2-1}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^4-2 x^3+x^2-1}}\right ) \]
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Rubi [F] time = 1.75, 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^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]
[Out]
Defer[Int][Sqrt[-1 + x^2 - 2*x^3 + x^4]/(2 + x^2 + 4*x^3 - 2*x^4), x] + 3*Defer[Int][(x*Sqrt[-1 + x^2 - 2*x^3
+ x^4])/(-1 - 2*x^3 + x^4), x] - 2*Defer[Int][(x^2*Sqrt[-1 + x^2 - 2*x^3 + x^4])/(-1 - 2*x^3 + x^4), x] - 6*De
fer[Int][(x*Sqrt[-1 + x^2 - 2*x^3 + x^4])/(-2 - x^2 - 4*x^3 + 2*x^4), x] + 4*Defer[Int][(x^2*Sqrt[-1 + x^2 - 2
*x^3 + x^4])/(-2 - x^2 - 4*x^3 + 2*x^4), x]
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx &=\int \left (-\frac {x (-3+2 x) \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4}+\frac {\left (-1-6 x+4 x^2\right ) \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4}\right ) \, dx\\ &=-\int \frac {x (-3+2 x) \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4} \, dx+\int \frac {\left (-1-6 x+4 x^2\right ) \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4} \, dx\\ &=-\int \left (-\frac {3 x \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4}+\frac {2 x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4}\right ) \, dx+\int \left (\frac {\sqrt {-1+x^2-2 x^3+x^4}}{2+x^2+4 x^3-2 x^4}-\frac {6 x \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4}+\frac {4 x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4}\right ) \, dx\\ &=-\left (2 \int \frac {x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4} \, dx\right )+3 \int \frac {x \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4} \, dx+4 \int \frac {x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4} \, dx-6 \int \frac {x \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4} \, dx+\int \frac {\sqrt {-1+x^2-2 x^3+x^4}}{2+x^2+4 x^3-2 x^4} \, dx\\ \end {align*}
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Mathematica [C] time = 6.68, size = 59573, normalized size = 1045.14 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]
[Out]
Result too large to show
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IntegrateAlgebraic [A] time = 0.43, size = 57, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-2 x^3+x^2-1}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^4-2 x^3+x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^
4)),x]
[Out]
ArcTanh[x/Sqrt[-1 + x^2 - 2*x^3 + x^4]] - Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[-1 + x^2 - 2*x^3 + x^4]]
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fricas [B] time = 0.52, size = 180, normalized size = 3.16 \begin {gather*} \frac {1}{8} \, \sqrt {3} \sqrt {2} \log \left (-\frac {4 \, x^{8} - 16 \, x^{7} + 60 \, x^{6} - 88 \, x^{5} + 41 \, x^{4} + 16 \, x^{3} - 4 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{5} - 4 \, x^{4} + 5 \, x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1} - 44 \, x^{2} + 4}{4 \, x^{8} - 16 \, x^{7} + 12 \, x^{6} + 8 \, x^{5} - 7 \, x^{4} + 16 \, x^{3} + 4 \, x^{2} + 4}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - 2 \, x^{3} + 2 \, x^{2} + 2 \, \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1} x - 1}{x^{4} - 2 \, x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x, algorithm="fricas")
[Out]
1/8*sqrt(3)*sqrt(2)*log(-(4*x^8 - 16*x^7 + 60*x^6 - 88*x^5 + 41*x^4 + 16*x^3 - 4*sqrt(3)*sqrt(2)*(2*x^5 - 4*x^
4 + 5*x^3 - 2*x)*sqrt(x^4 - 2*x^3 + x^2 - 1) - 44*x^2 + 4)/(4*x^8 - 16*x^7 + 12*x^6 + 8*x^5 - 7*x^4 + 16*x^3 +
4*x^2 + 4)) + 1/2*log(-(x^4 - 2*x^3 + 2*x^2 + 2*sqrt(x^4 - 2*x^3 + x^2 - 1)*x - 1)/(x^4 - 2*x^3 - 1))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + 1\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1}}{{\left (2 \, x^{4} - 4 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - 2 \, x^{3} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x, algorithm="giac")
[Out]
integrate((x^4 - x^3 + 1)*sqrt(x^4 - 2*x^3 + x^2 - 1)/((2*x^4 - 4*x^3 - x^2 - 2)*(x^4 - 2*x^3 - 1)), x)
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maple [C] time = 1.02, size = 1574, normalized size = 27.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x)
[Out]
-1/3*I*(-1/2*I*3^(1/2)-1/2*5^(1/2))*((1/2*5^(1/2)-1/2*I*3^(1/2))*(x-1/2+1/2*I*3^(1/2))/(1/2*5^(1/2)+1/2*I*3^(1
/2))/(x-1/2-1/2*I*3^(1/2)))^(1/2)*(x-1/2-1/2*I*3^(1/2))^2*(I*3^(1/2)*(x-1/2+1/2*5^(1/2))/(-1/2*5^(1/2)+1/2*I*3
^(1/2))/(x-1/2-1/2*I*3^(1/2)))^(1/2)*(I*3^(1/2)*(x-1/2-1/2*5^(1/2))/(1/2*5^(1/2)+1/2*I*3^(1/2))/(x-1/2-1/2*I*3
^(1/2)))^(1/2)/(1/2*5^(1/2)-1/2*I*3^(1/2))*3^(1/2)/((x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2*I*3^(1/2))*(x-1/2+1/2*5^(
1/2))*(x-1/2-1/2*5^(1/2)))^(1/2)*EllipticF(((1/2*5^(1/2)-1/2*I*3^(1/2))*(x-1/2+1/2*I*3^(1/2))/(1/2*5^(1/2)+1/2
*I*3^(1/2))/(x-1/2-1/2*I*3^(1/2)))^(1/2),((1/2*5^(1/2)+1/2*I*3^(1/2))*(-1/2*I*3^(1/2)-1/2*5^(1/2))/(1/2*5^(1/2
)-1/2*I*3^(1/2))/(-1/2*5^(1/2)+1/2*I*3^(1/2)))^(1/2))-1/96*I*16^(1/2)*sum(_alpha*(-I*3^(1/2)-5^(1/2))*((5^(1/2
)-I*3^(1/2))*(-1+2*x+I*3^(1/2))/(5^(1/2)+I*3^(1/2))/(-1+2*x-I*3^(1/2)))^(1/2)*(-1+2*x-I*3^(1/2))^2*(2*I*(2*x-1
+5^(1/2))/(I*3^(1/2)-5^(1/2))/(-1+2*x-I*3^(1/2)))^(1/2)*(2*I*(-1+2*x-5^(1/2))/(5^(1/2)+I*3^(1/2))/(-1+2*x-I*3^
(1/2)))^(1/2)/(5^(1/2)-I*3^(1/2))/((-1+2*x+I*3^(1/2))*(-1+2*x-I*3^(1/2))*(2*x-1+5^(1/2))*(-1+2*x-5^(1/2)))^(1/
2)*(2*_alpha^3-6*_alpha^2-_alpha+7+I*3^(1/2)*(2*_alpha^3-2*_alpha^2-5*_alpha-3))*(6*EllipticF(((1/2*5^(1/2)-1/
2*I*3^(1/2))*(x-1/2+1/2*I*3^(1/2))/(1/2*5^(1/2)+1/2*I*3^(1/2))/(x-1/2-1/2*I*3^(1/2)))^(1/2),((1/2*5^(1/2)+1/2*
I*3^(1/2))*(-1/2*I*3^(1/2)-1/2*5^(1/2))/(1/2*5^(1/2)-1/2*I*3^(1/2))/(-1/2*5^(1/2)+1/2*I*3^(1/2)))^(1/2))+I*3^(
1/2)*(2*_alpha^3-6*_alpha^2-_alpha+7+I*3^(1/2)*(-2*_alpha^3+2*_alpha^2+5*_alpha+3))*EllipticPi(((1/2*5^(1/2)-1
/2*I*3^(1/2))*(x-1/2+1/2*I*3^(1/2))/(1/2*5^(1/2)+1/2*I*3^(1/2))/(x-1/2-1/2*I*3^(1/2)))^(1/2),-1/8-1/4*_alpha^2
+1/4*_alpha^3+1/4*I*5^(1/2)*3^(1/2)*_alpha^3-5/8*_alpha-1/24*I*3^(1/2)*_alpha-1/8*I*5^(1/2)*3^(1/2)+1/12*I*3^(
1/2)*_alpha^3-1/4*5^(1/2)*_alpha^3+3/4*5^(1/2)*_alpha^2+1/8*_alpha*5^(1/2)-1/4*I*3^(1/2)*_alpha^2+7/24*I*3^(1/
2)-1/4*I*5^(1/2)*3^(1/2)*_alpha^2-5/8*I*5^(1/2)*3^(1/2)*_alpha-7/8*5^(1/2),((1/2*5^(1/2)+1/2*I*3^(1/2))*(-1/2*
I*3^(1/2)-1/2*5^(1/2))/(1/2*5^(1/2)-1/2*I*3^(1/2))/(-1/2*5^(1/2)+1/2*I*3^(1/2)))^(1/2))),_alpha=RootOf(2*_Z^4-
4*_Z^3-_Z^2-2))+1/16*I*16^(1/2)*sum(_alpha*(-I*3^(1/2)-5^(1/2))*((5^(1/2)-I*3^(1/2))*(-1+2*x+I*3^(1/2))/(5^(1/
2)+I*3^(1/2))/(-1+2*x-I*3^(1/2)))^(1/2)*(-1+2*x-I*3^(1/2))^2*(2*I*(2*x-1+5^(1/2))/(I*3^(1/2)-5^(1/2))/(-1+2*x-
I*3^(1/2)))^(1/2)*(2*I*(-1+2*x-5^(1/2))/(5^(1/2)+I*3^(1/2))/(-1+2*x-I*3^(1/2)))^(1/2)/(5^(1/2)-I*3^(1/2))/((-1
+2*x+I*3^(1/2))*(-1+2*x-I*3^(1/2))*(2*x-1+5^(1/2))*(-1+2*x-5^(1/2)))^(1/2)*(_alpha^3-3*_alpha^2+3+I*3^(1/2)*(_
alpha^3-_alpha^2-2*_alpha-1))*(2*EllipticF(((1/2*5^(1/2)-1/2*I*3^(1/2))*(x-1/2+1/2*I*3^(1/2))/(1/2*5^(1/2)+1/2
*I*3^(1/2))/(x-1/2-1/2*I*3^(1/2)))^(1/2),((1/2*5^(1/2)+1/2*I*3^(1/2))*(-1/2*I*3^(1/2)-1/2*5^(1/2))/(1/2*5^(1/2
)-1/2*I*3^(1/2))/(-1/2*5^(1/2)+1/2*I*3^(1/2)))^(1/2))+I*3^(1/2)*(_alpha^3-3*_alpha^2+3+I*3^(1/2)*(-_alpha^3+_a
lpha^2+2*_alpha+1))*EllipticPi(((1/2*5^(1/2)-1/2*I*3^(1/2))*(x-1/2+1/2*I*3^(1/2))/(1/2*5^(1/2)+1/2*I*3^(1/2))/
(x-1/2-1/2*I*3^(1/2)))^(1/2),1/8*I*_alpha^3*3^(1/2)-9/8*5^(1/2)-3/8*5^(1/2)*_alpha^3+3/8*I*3^(1/2)+9/8*5^(1/2)
*_alpha^2-3/4*_alpha-3/8*I*_alpha^2*3^(1/2)+3/8*I*_alpha^3*5^(1/2)*3^(1/2)-1/8*I*5^(1/2)*3^(1/2)-3/8*I*_alpha^
2*5^(1/2)*3^(1/2)-3/4*I*_alpha*5^(1/2)*3^(1/2)+3/8*_alpha^3-1/8-3/8*_alpha^2,((1/2*5^(1/2)+1/2*I*3^(1/2))*(-1/
2*I*3^(1/2)-1/2*5^(1/2))/(1/2*5^(1/2)-1/2*I*3^(1/2))/(-1/2*5^(1/2)+1/2*I*3^(1/2)))^(1/2))),_alpha=RootOf(_Z^4-
2*_Z^3-1))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + 1\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1}}{{\left (2 \, x^{4} - 4 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - 2 \, x^{3} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x, algorithm="maxima")
[Out]
integrate((x^4 - x^3 + 1)*sqrt(x^4 - 2*x^3 + x^2 - 1)/((2*x^4 - 4*x^3 - x^2 - 2)*(x^4 - 2*x^3 - 1)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^4-x^3+1\right )\,\sqrt {x^4-2\,x^3+x^2-1}}{\left (-x^4+2\,x^3+1\right )\,\left (-2\,x^4+4\,x^3+x^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^4 - x^3 + 1)*(x^2 - 2*x^3 + x^4 - 1)^(1/2))/((2*x^3 - x^4 + 1)*(x^2 + 4*x^3 - 2*x^4 + 2)),x)
[Out]
int(((x^4 - x^3 + 1)*(x^2 - 2*x^3 + x^4 - 1)^(1/2))/((2*x^3 - x^4 + 1)*(x^2 + 4*x^3 - 2*x^4 + 2)), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**4-2*x**3+x**2-1)**(1/2)*(x**4-x**3+1)/(x**4-2*x**3-1)/(2*x**4-4*x**3-x**2-2),x)
[Out]
Timed out
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