\(\int \frac {-1-2 x+2 x^2}{(1+2 x^2) \sqrt {x+x^4}} \, dx\) [161]

Optimal result

Integrand size = 29, antiderivative size = 19 \[ \int \frac {-1-2 x+2 x^2}{\left (1+2 x^2\right ) \sqrt {x+x^4}} \, dx=\arctan \left (\frac {2 \sqrt {x+x^4}}{-1+2 x}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(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])/(3^(1/4)*Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4]) - ((1/2 + I/2)*(I
+ Sqrt[2])*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/(((-1)^(1/4) - 2^(1/4)*x)*Sqrt[1 + x^6]), x], x, Sq
rt[x]])/Sqrt[x + x^4] + ((1/2 + I/2)*(1 + I*Sqrt[2])*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((-(-1)^(
3/4) - 2^(1/4)*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/Sqrt[x + x^4] - ((1/2 + I/2)*(I + Sqrt[2])*Sqrt[x]*Sqrt[1 +
x^3]*Defer[Subst][Defer[Int][1/(((-1)^(1/4) + 2^(1/4)*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/Sqrt[x + x^4] + ((1
/2 + I/2)*(1 + I*Sqrt[2])*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((-(-1)^(3/4) + 2^(1/4)*x)*Sqrt[1 +
x^6]), x], x, Sqrt[x]])/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} \left (1+2 x^2\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {1+x^3}}-\frac {2 (1+x)}{\sqrt {x} \left (1+2 x^2\right ) \sqrt {1+x^3}}\right ) \, dx}{\sqrt {x+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}-\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1+x}{\sqrt {x} \left (1+2 x^2\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = -\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \int \left (\frac {i-\frac {1}{\sqrt {2}}}{2 \sqrt {x} \left (i-\sqrt {2} x\right ) \sqrt {1+x^3}}+\frac {i+\frac {1}{\sqrt {2}}}{2 \sqrt {x} \left (i+\sqrt {2} x\right ) \sqrt {1+x^3}}\right ) \, dx}{\sqrt {x+x^4}}+\frac {\left (2 \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 {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 )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (\left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (i-\sqrt {2} x\right ) \sqrt {1+x^3}} \, dx}{2 \sqrt {x+x^4}}-\frac {\left (\left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (i+\sqrt {2} x\right ) \sqrt {1+x^3}} \, dx}{2 \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 )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (\left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i-\sqrt {2} x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (\left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {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 )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (\left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (\left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}}\right ) \, 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 )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}+\frac {\left ((-1)^{3/4} \left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-1}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}}+\frac {\left ((-1)^{3/4} \left (2 i-\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-1}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}}+\frac {\left (\sqrt [4]{-1} \left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{3/4}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}}+\frac {\left (\sqrt [4]{-1} \left (2 i+\sqrt {2}\right ) \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{3/4}+\sqrt [4]{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. \(41\) vs. \(2(19)=38\).

Time = 21.95 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.16 \[ \int \frac {-1-2 x+2 x^2}{\left (1+2 x^2\right ) \sqrt {x+x^4}} \, dx=\frac {2 \sqrt {1+\frac {1}{x^3}} x^2 \arctan \left (\frac {\sqrt {1+\frac {1}{x^3}}}{1+\frac {1}{x}}\right )}{\sqrt {x+x^4}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.66 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{4}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{2}+1}\right )\)	\(46\)
default	\(\text {Expression too large to display}\)	\(13532\)
elliptic	\(\text {Expression too large to display}\)	\(14832\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-1-2 x+2 x^2}{\left (1+2 x^2\right ) \sqrt {x+x^4}} \, dx=-\arctan \left (\frac {2 \, x - 1}{2 \, \sqrt {x^{4} + x}}\right ) \]

[In]

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

[Out]

-arctan(1/2*(2*x - 1)/sqrt(x^4 + x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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