\(\int \frac {(-1+2 x) (2-x+x^2) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx\) [1286]

Optimal result

Integrand size = 43, antiderivative size = 93 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \left (1-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}-\frac {1}{4} \text {arctanh}\left (3-2 x+2 x^2+2 \sqrt {-2+x^2-2 x^3+x^4}\right )-\frac {7}{8} \log \left (-x+x^2+\sqrt {-2+x^2-2 x^3+x^4}\right ) \]

[Out]

1/4*(x^2-x+1)*(x^4-2*x^3+x^2-2)^(1/2)-1/4*arctanh(3-2*x+2*x^2+2*(x^4-2*x^3+x^2-2)^(1/2))-7/8*ln(-x+x^2+(x^4-2*
x^3+x^2-2)^(1/2))

Rubi [F]

\[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx \]

[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*} \text {integral}& = \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} \text {Subst}\left (\int \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )+\text {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} \text {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+\text {Subst}\left (\int \frac {1}{2} \sqrt {-\frac {31}{16}-\frac {x^2}{2}+x^4} \, dx,x,-\frac {1}{2}+x\right )+\text {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} \text {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} \text {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} \text {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 ) \text {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 (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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} \text {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} \text {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 (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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} \text {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 ) \text {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-\text {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} \text {arctanh}\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*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \left (1-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}+\frac {1}{4} \text {arctanh}\left (3-2 x+2 x^2-2 \sqrt {-2+x^2-2 x^3+x^4}\right )+\frac {7}{8} \log \left (x-x^2+\sqrt {-2+x^2-2 x^3+x^4}\right ) \]

[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]

((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

Maple [A] (verified)

Time = 10.67 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87

method	result	size
risch	\(\frac {\left (x^{2}-x +1\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}}{4}-\frac {7 \ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}-2}\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {3 x^{2}-3 x +4}{\sqrt {x^{4}-2 x^{3}+x^{2}-2}}\right )}{8}\)	\(81\)
default	\(\frac {\operatorname {arctanh}\left (\frac {3 x^{2}-3 x +4}{\sqrt {x^{4}-2 x^{3}+x^{2}-2}}\right )}{8}-\frac {7 \ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}-2}\right )}{8}+\frac {\left (2 x^{2}-2 x +2\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}}{8}\)	\(83\)
pseudoelliptic	\(\frac {\operatorname {arctanh}\left (\frac {3 x^{2}-3 x +4}{\sqrt {x^{4}-2 x^{3}+x^{2}-2}}\right )}{8}-\frac {7 \ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}-2}\right )}{8}+\frac {\left (2 x^{2}-2 x +2\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}}{8}\)	\(83\)
trager	\(\left (\frac {1}{4} x^{2}-\frac {1}{4} x +\frac {1}{4}\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}+\frac {\ln \left (-\frac {1-2 x \sqrt {x^{4}-2 x^{3}+x^{2}-2}+8 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{14}-56 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{13}+184 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{12}-376 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{11}+504 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{10}-408 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{9}+96 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{8}+216 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{7}+26 x^{2} \sqrt {x^{4}-2 x^{3}+x^{2}-2}-14 x -186 x^{4}-624 x^{7}+176 x^{5}+12 x^{2}+60 x^{3}+784 x^{8}+152 x^{6}-368 x^{10}+864 x^{11}-872 x^{12}+560 x^{13}-240 x^{14}+64 x^{15}-8 x^{16}-360 x^{9}-2 \sqrt {x^{4}-2 x^{3}+x^{2}-2}-300 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{6}+164 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{5}-4 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{4}-52 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{3}}{2 x^{2}-2 x +3}\right )}{8}\)	\(420\)
elliptic	\(\text {Expression too large to display}\)	\(3475\)

[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*(x^2-x+1)*(x^4-2*x^3+x^2-2)^(1/2)-7/8*ln(-x+x^2+(x^4-2*x^3+x^2-2)^(1/2))+1/8*arctanh((3*x^2-3*x+4)/(x^4-2*
x^3+x^2-2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \, \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} {\left (x^{2} - x + 1\right )} + \frac {7}{8} \, \log \left (-x^{2} + x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2}\right ) + \frac {1}{8} \, \log \left (\frac {3 \, x^{2} - 3 \, x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} + 4}{2 \, x^{2} - 2 \, x + 3}\right ) \]

[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))

Sympy [F]

\[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int \frac {\left (2 x - 1\right ) \left (x^{2} - x + 2\right ) \sqrt {x^{4} - 2 x^{3} + x^{2} - 2}}{2 x^{2} - 2 x + 3}\, dx \]

[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)

Maxima [F]

\[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int { \frac {\sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} {\left (x^{2} - x + 2\right )} {\left (2 \, x - 1\right )}}{2 \, x^{2} - 2 \, x + 3} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \, \sqrt {{\left (x^{2} - x\right )}^{2} - 2} {\left (x^{2} - x + 1\right )} + \frac {1}{8} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} - 2} + 2\right ) - \frac {1}{8} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} - 2} + 1\right ) + \frac {7}{8} \, \log \left ({\left | -x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} - 2} \right |}\right ) \]

[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)))

Mupad [F(-1)]

Timed out. \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int \frac {\left (2\,x-1\right )\,\left (x^2-x+2\right )\,\sqrt {x^4-2\,x^3+x^2-2}}{2\,x^2-2\,x+3} \,d x \]

[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)