3.13.86 \(\int \frac {(-1+2 x) (2-x+x^2) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx\)

Optimal. Leaf size=93 \[ \frac {1}{4} \sqrt {x^4-2 x^3+x^2-2} \left (x^2-x+1\right )-\frac {7}{8} \log \left (x^2+\sqrt {x^4-2 x^3+x^2-2}-x\right )-\frac {1}{4} \tanh ^{-1}\left (2 x^2+2 \sqrt {x^4-2 x^3+x^2-2}-2 x+3\right ) \]

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Rubi [F]  time = 1.83, 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+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/4*((1 - x)*x*Sqrt[-2 + x^2 - 2*x^3 + x^4]) + ArcTanh[((1 - x)*x)/Sqrt[-2 + x^2 - 2*x^3 + x^4]]/2 + Defer[In
t][Sqrt[-2 + x^2 - 2*x^3 + x^4]/(-2 - (2*I)*Sqrt[5] + 4*x), x] + Defer[Int][Sqrt[-2 + x^2 - 2*x^3 + x^4]/(-2 +
(2*I)*Sqrt[5] + 4*x), x]

Rubi steps

\begin {align*} \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx &=\int \left (-\frac {1}{2} \sqrt {-2+x^2-2 x^3+x^4}+x \sqrt {-2+x^2-2 x^3+x^4}-\frac {(1-2 x) \sqrt {-2+x^2-2 x^3+x^4}}{2 \left (3-2 x+2 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \sqrt {-2+x^2-2 x^3+x^4} \, dx\right )-\frac {1}{2} \int \frac {(1-2 x) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx+\int x \sqrt {-2+x^2-2 x^3+x^4} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {2 \sqrt {-2+x^2-2 x^3+x^4}}{-2-2 i \sqrt {5}+4 x}-\frac {2 \sqrt {-2+x^2-2 x^3+x^4}}{-2+2 i \sqrt {5}+4 x}\right ) \, dx\right )-\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )+\operatorname {Subst}\left (\int \left (\frac {1}{2}+x\right ) \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )\\ &=\frac {1}{12} (1-2 x) \sqrt {-2+x^2-2 x^3+x^4}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {-\frac {31}{8}-\frac {x^2}{2}}{\sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4}} \, dx,x,-\frac {1}{2}+x\right )+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2-2 i \sqrt {5}+4 x} \, dx+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2+2 i \sqrt {5}+4 x} \, dx+\operatorname {Subst}\left (\int \frac {1}{2} \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )+\operatorname {Subst}\left (\int x \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )\\ &=\frac {1}{12} (1-2 x) \sqrt {-2+x^2-2 x^3+x^4}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-2 \sqrt {2}+2 x^2}{\sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4}} \, dx,x,-\frac {1}{2}+x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {-\frac {31}{16}-\frac {x}{2}+x^2} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )+\frac {1}{12} \left (8+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4}} \, dx,x,-\frac {1}{2}+x\right )+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2-2 i \sqrt {5}+4 x} \, dx+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2+2 i \sqrt {5}+4 x} \, dx\\ &=-\frac {(1-2 x) \left (\sqrt {2}-x+x^2\right )}{24 \sqrt {-2+x^2-2 x^3+x^4}}-\frac {1}{4} (1-x) x \sqrt {-2+x^2-2 x^3+x^4}+\frac {\sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2} \sqrt {\frac {31+\left (1+4 \sqrt {2}\right ) (1-2 x)^2}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} E\left (\sin ^{-1}\left (\frac {2\ 2^{3/4} (1-2 x)}{\sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2}}\right )|\frac {1}{16} \left (8-\sqrt {2}\right )\right )}{24 \sqrt [4]{2} \sqrt {31} \sqrt {\frac {1}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} \sqrt {-2+x^2-2 x^3+x^4}}-\frac {\left (1+4 \sqrt {2}\right ) \sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2} \sqrt {\frac {31+\left (1+4 \sqrt {2}\right ) (1-2 x)^2}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} F\left (\sin ^{-1}\left (\frac {2\ 2^{3/4} (1-2 x)}{\sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2}}\right )|\frac {1}{16} \left (8-\sqrt {2}\right )\right )}{48 \sqrt [4]{2} \sqrt {31} \sqrt {\frac {1}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} \sqrt {-2+x^2-2 x^3+x^4}}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {-\frac {31}{8}-\frac {x^2}{2}}{\sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4}} \, dx,x,-\frac {1}{2}+x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {31}{16}-\frac {x}{2}+x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2-2 i \sqrt {5}+4 x} \, dx+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2+2 i \sqrt {5}+4 x} \, dx\\ &=-\frac {(1-2 x) \left (\sqrt {2}-x+x^2\right )}{24 \sqrt {-2+x^2-2 x^3+x^4}}-\frac {1}{4} (1-x) x \sqrt {-2+x^2-2 x^3+x^4}+\frac {\sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2} \sqrt {\frac {31+\left (1+4 \sqrt {2}\right ) (1-2 x)^2}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} E\left (\sin ^{-1}\left (\frac {2\ 2^{3/4} (1-2 x)}{\sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2}}\right )|\frac {1}{16} \left (8-\sqrt {2}\right )\right )}{24 \sqrt [4]{2} \sqrt {31} \sqrt {\frac {1}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} \sqrt {-2+x^2-2 x^3+x^4}}-\frac {\left (1+4 \sqrt {2}\right ) \sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2} \sqrt {\frac {31+\left (1+4 \sqrt {2}\right ) (1-2 x)^2}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} F\left (\sin ^{-1}\left (\frac {2\ 2^{3/4} (1-2 x)}{\sqrt {-31-\left (1-4 \sqrt {2}\right ) (1-2 x)^2}}\right )|\frac {1}{16} \left (8-\sqrt {2}\right )\right )}{48 \sqrt [4]{2} \sqrt {31} \sqrt {\frac {1}{31+\left (1-4 \sqrt {2}\right ) (1-2 x)^2}} \sqrt {-2+x^2-2 x^3+x^4}}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-2 \sqrt {2}+2 x^2}{\sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4}} \, dx,x,-\frac {1}{2}+x\right )-\frac {1}{12} \left (8+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4}} \, dx,x,-\frac {1}{2}+x\right )+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2-2 i \sqrt {5}+4 x} \, dx+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2+2 i \sqrt {5}+4 x} \, dx-\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2 (-1+x) x}{\sqrt {-2+x^2-2 x^3+x^4}}\right )\\ &=-\frac {1}{4} (1-x) x \sqrt {-2+x^2-2 x^3+x^4}-\frac {1}{2} \tanh ^{-1}\left (\frac {(-1+x) x}{\sqrt {-2+x^2-2 x^3+x^4}}\right )+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2-2 i \sqrt {5}+4 x} \, dx+\int \frac {\sqrt {-2+x^2-2 x^3+x^4}}{-2+2 i \sqrt {5}+4 x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 88, normalized size = 0.95 \begin {gather*} \frac {1}{8} \left (2 \sqrt {x^4-2 x^3+x^2-2} \left (x^2-x+1\right )-7 \tanh ^{-1}\left (\frac {(x-1) x}{\sqrt {x^4-2 x^3+x^2-2}}\right )-\tanh ^{-1}\left (\frac {-3 x^2+3 x-4}{\sqrt {x^4-2 x^3+x^2-2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - x + x^2)*Sqrt[-2 + x^2 - 2*x^3 + x^4] - 7*ArcTanh[((-1 + x)*x)/Sqrt[-2 + x^2 - 2*x^3 + x^4]] - ArcTanh
[(-4 + 3*x - 3*x^2)/Sqrt[-2 + x^2 - 2*x^3 + x^4]])/8

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IntegrateAlgebraic [A]  time = 0.30, size = 93, normalized size = 1.00 \begin {gather*} \frac {1}{4} \left (1-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}-\frac {1}{4} \tanh ^{-1}\left (3-2 x+2 x^2+2 \sqrt {-2+x^2-2 x^3+x^4}\right )-\frac {7}{8} \log \left (-x+x^2+\sqrt {-2+x^2-2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - x + x^2)*Sqrt[-2 + x^2 - 2*x^3 + x^4])/4 - ArcTanh[3 - 2*x + 2*x^2 + 2*Sqrt[-2 + x^2 - 2*x^3 + x^4]]/4 -
(7*Log[-x + x^2 + Sqrt[-2 + x^2 - 2*x^3 + x^4]])/8

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fricas [A]  time = 0.55, size = 92, normalized size = 0.99 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} {\left (x^{2} - x + 1\right )} + \frac {7}{8} \, \log \left (-x^{2} + x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2}\right ) + \frac {1}{8} \, \log \left (\frac {3 \, x^{2} - 3 \, x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} + 4}{2 \, x^{2} - 2 \, x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*(x^2-x+2)*(x^4-2*x^3+x^2-2)^(1/2)/(2*x^2-2*x+3),x, algorithm="fricas")

[Out]

1/4*sqrt(x^4 - 2*x^3 + x^2 - 2)*(x^2 - x + 1) + 7/8*log(-x^2 + x + sqrt(x^4 - 2*x^3 + x^2 - 2)) + 1/8*log((3*x
^2 - 3*x + sqrt(x^4 - 2*x^3 + x^2 - 2) + 4)/(2*x^2 - 2*x + 3))

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giac [A]  time = 0.35, size = 100, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, \sqrt {{\left (x^{2} - x\right )}^{2} - 2} {\left (x^{2} - x + 1\right )} + \frac {1}{8} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} - 2} + 2\right ) - \frac {1}{8} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} - 2} + 1\right ) + \frac {7}{8} \, \log \left ({\left | -x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} - 2} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*(x^2-x+2)*(x^4-2*x^3+x^2-2)^(1/2)/(2*x^2-2*x+3),x, algorithm="giac")

[Out]

1/4*sqrt((x^2 - x)^2 - 2)*(x^2 - x + 1) + 1/8*log(x^2 - x - sqrt((x^2 - x)^2 - 2) + 2) - 1/8*log(x^2 - x - sqr
t((x^2 - x)^2 - 2) + 1) + 7/8*log(abs(-x^2 + x + sqrt((x^2 - x)^2 - 2)))

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maple [B]  time = 2.70, size = 420, normalized size = 4.52

method	result	size
trager	\(\left (\frac {1}{4}-\frac {1}{4} x +\frac {1}{4} x^{2}\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}-\frac {\ln \left (-\frac {-1+14 x +26 x^{2} \sqrt {x^{4}-2 x^{3}+x^{2}-2}-2 x \sqrt {x^{4}-2 x^{3}+x^{2}-2}-2 \sqrt {x^{4}-2 x^{3}+x^{2}-2}+360 x^{9}+368 x^{10}+624 x^{7}-176 x^{5}-152 x^{6}-12 x^{2}-60 x^{3}-784 x^{8}+186 x^{4}-864 x^{11}+872 x^{12}+8 x^{16}-64 x^{15}+240 x^{14}-560 x^{13}+8 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{14}-56 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{13}+184 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{12}-376 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{11}+504 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{10}-408 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{9}+96 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{8}+216 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{7}-300 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{6}+164 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{5}-4 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{4}-52 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{3}}{2 x^{2}-2 x +3}\right )}{8}\)	\(420\)
risch	\(\text {Expression too large to display}\)	\(3445\)
default	\(\text {Expression too large to display}\)	\(3475\)
elliptic	\(\text {Expression too large to display}\)	\(3475\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+2*x)*(x^2-x+2)*(x^4-2*x^3+x^2-2)^(1/2)/(2*x^2-2*x+3),x,method=_RETURNVERBOSE)

[Out]

(1/4-1/4*x+1/4*x^2)*(x^4-2*x^3+x^2-2)^(1/2)-1/8*ln(-(-1+14*x+26*x^2*(x^4-2*x^3+x^2-2)^(1/2)-2*x*(x^4-2*x^3+x^2
-2)^(1/2)-2*(x^4-2*x^3+x^2-2)^(1/2)+360*x^9+368*x^10+624*x^7-176*x^5-152*x^6-12*x^2-60*x^3-784*x^8+186*x^4-864
*x^11+872*x^12+8*x^16-64*x^15+240*x^14-560*x^13+8*(x^4-2*x^3+x^2-2)^(1/2)*x^14-56*(x^4-2*x^3+x^2-2)^(1/2)*x^13
+184*(x^4-2*x^3+x^2-2)^(1/2)*x^12-376*(x^4-2*x^3+x^2-2)^(1/2)*x^11+504*(x^4-2*x^3+x^2-2)^(1/2)*x^10-408*(x^4-2
*x^3+x^2-2)^(1/2)*x^9+96*(x^4-2*x^3+x^2-2)^(1/2)*x^8+216*(x^4-2*x^3+x^2-2)^(1/2)*x^7-300*(x^4-2*x^3+x^2-2)^(1/
2)*x^6+164*(x^4-2*x^3+x^2-2)^(1/2)*x^5-4*(x^4-2*x^3+x^2-2)^(1/2)*x^4-52*(x^4-2*x^3+x^2-2)^(1/2)*x^3)/(2*x^2-2*
x+3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} {\left (x^{2} - x + 2\right )} {\left (2 \, x - 1\right )}}{2 \, x^{2} - 2 \, x + 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*(x^2-x+2)*(x^4-2*x^3+x^2-2)^(1/2)/(2*x^2-2*x+3),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - 2*x^3 + x^2 - 2)*(x^2 - x + 2)*(2*x - 1)/(2*x^2 - 2*x + 3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x-1\right )\,\left (x^2-x+2\right )\,\sqrt {x^4-2\,x^3+x^2-2}}{2\,x^2-2\,x+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)*(x^2 - x + 2)*(x^2 - 2*x^3 + x^4 - 2)^(1/2))/(2*x^2 - 2*x + 3),x)

[Out]

int(((2*x - 1)*(x^2 - x + 2)*(x^2 - 2*x^3 + x^4 - 2)^(1/2))/(2*x^2 - 2*x + 3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x - 1\right ) \left (x^{2} - x + 2\right ) \sqrt {x^{4} - 2 x^{3} + x^{2} - 2}}{2 x^{2} - 2 x + 3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)*(x**2-x+2)*(x**4-2*x**3+x**2-2)**(1/2)/(2*x**2-2*x+3),x)

[Out]

Integral((2*x - 1)*(x**2 - x + 2)*sqrt(x**4 - 2*x**3 + x**2 - 2)/(2*x**2 - 2*x + 3), x)

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