\(\int \frac {-1-2 x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx\) [196]

Optimal result

Integrand size = 27, antiderivative size = 21 \[ \int \frac {-1-2 x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\text {arctanh}\left (\frac {2 \sqrt {x+x^4}}{1+2 x^2}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(2*Sqrt[x]*Sqrt[1 + x^3]*ArcSinh[x^(3/2)])/(3*Sqrt[x + x^4]) - (x*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3]
)*x)^2]*EllipticF[ArcCos[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3])*x)], (2 + Sqrt[3])/4])/(2*3^(1/4)*Sqrt[(x*(1
+ x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4]) + (3*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((1 - Sqrt
[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[x + x^4]) + (3*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1
/((1 + Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[x + x^4])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {-1-2 x+2 x^2}{\sqrt {x} (-1+2 x) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {-1-2 x^2+2 x^4}{\left (-1+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 \sqrt {1+x^6}}+\frac {x^2}{\sqrt {1+x^6}}-\frac {3}{2 \left (-1+2 x^2\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = -\frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = -\frac {x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 \sqrt {x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 \left (1-\sqrt {2} x\right ) \sqrt {1+x^6}}-\frac {1}{2 \left (1+\sqrt {2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {2 \sqrt {x} \sqrt {1+x^3} \text {arcsinh}\left (x^{3/2}\right )}{3 \sqrt {x+x^4}}-\frac {x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(21)=42\).

Time = 7.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {-1-2 x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {2 \sqrt {x} \sqrt {1+x^3} \text {arctanh}\left (\frac {\sqrt {x} \sqrt {1+x^3}}{1-x+x^2}\right )}{\sqrt {x+x^4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33

method	result	size
trager	\(-\ln \left (\frac {-2 x^{2}+2 \sqrt {x^{4}+x}-1}{-1+2 x}\right )\)	\(28\)
default	\(\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(511\)
elliptic	\(\text {Expression too large to display}\)	\(780\)

[In]

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

[Out]

-ln((-2*x^2+2*(x^4+x)^(1/2)-1)/(-1+2*x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-1-2 x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\log \left (\frac {2 \, x^{2} + 2 \, \sqrt {x^{4} + x} + 1}{2 \, x - 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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