\(\int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx\) [880]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {(-1+x) \sqrt {1-4 x+3 x^2+2 x^3+x^4}}{x^3}\right )+\text {arctanh}\left (\frac {-1+2 x+\sqrt {1-4 x+3 x^2+2 x^3+x^4}}{x^2}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {x^3}{(-1+x) \sqrt {1-4 x+3 x^2+2 x^3+x^4}}\right )-\text {arctanh}\left (\frac {-\sqrt {3}+2 \sqrt {3} x-\sqrt {1-4 x+3 x^2+2 x^3+x^4}}{(-1+x)^2}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(61)=122\).

Time = 1.42 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.25

method	result	size
default	\(\frac {\ln \left (2 x^{6}+2 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{4}+12 x^{5}+10 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{3}+36 x^{4}+24 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{2}+56 x^{3}+28 x \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}+42 x^{2}+14 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}-13\right )}{6}\)	\(151\)
trager	\(\frac {\ln \left (2 x^{6}+2 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{4}+12 x^{5}+10 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{3}+36 x^{4}+24 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{2}+56 x^{3}+28 x \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}+42 x^{2}+14 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}-13\right )}{6}\)	\(151\)
elliptic	\(\text {Expression too large to display}\)	\(1609\)

[In]

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

[Out]

1/6*ln(2*x^6+2*(x^4+2*x^3+3*x^2-4*x+1)^(1/2)*x^4+12*x^5+10*(x^4+2*x^3+3*x^2-4*x+1)^(1/2)*x^3+36*x^4+24*(x^4+2*
x^3+3*x^2-4*x+1)^(1/2)*x^2+56*x^3+28*x*(x^4+2*x^3+3*x^2-4*x+1)^(1/2)+42*x^2+14*(x^4+2*x^3+3*x^2-4*x+1)^(1/2)-1
3)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx=\frac {1}{6} \, \log \left (2 \, x^{6} + 12 \, x^{5} + 36 \, x^{4} + 56 \, x^{3} + 42 \, x^{2} + 2 \, {\left (x^{4} + 5 \, x^{3} + 12 \, x^{2} + 14 \, x + 7\right )} \sqrt {x^{4} + 2 \, x^{3} + 3 \, x^{2} - 4 \, x + 1} - 13\right ) \]

[In]

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

[Out]

1/6*log(2*x^6 + 12*x^5 + 36*x^4 + 56*x^3 + 42*x^2 + 2*(x^4 + 5*x^3 + 12*x^2 + 14*x + 7)*sqrt(x^4 + 2*x^3 + 3*x
^2 - 4*x + 1) - 13)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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