\(\int \frac {-2+2 x+x^2}{(2-4 x+3 x^2) \sqrt {1+x^3}} \, dx\) [397]

Optimal result

Integrand size = 30, antiderivative size = 33 \[ \int \frac {-2+2 x+x^2}{\left (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {3} \left (1-x+x^2\right )}\right )}{\sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2170, 212} \[ \int \frac {-2+2 x+x^2}{\left (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {x+1}{\sqrt {3} \sqrt {x^3+1}}\right )}{\sqrt {3}} \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2170

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /;
FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h,
0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]

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

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+x^2}{\left (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {3} \left (1-x+x^2\right )}\right )}{\sqrt {3}} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 6.72 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{3}+1}}{3 x^{2}-4 x +2}\right )}{3}\)	\(62\)
default	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {4 \left (\frac {5}{6}+\frac {i \sqrt {2}}{6}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}-\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \sqrt {3}\, \left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {4 \left (\frac {5}{6}-\frac {i \sqrt {2}}{6}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}+\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \sqrt {3}\, \left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\)	\(435\)
elliptic	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}-\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \sqrt {3}\, \left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}+\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \sqrt {3}\, \left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\)	\(435\)

[In]

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

[Out]

-1/3*RootOf(_Z^2-3)*ln(-(3*RootOf(_Z^2-3)*x^2-2*RootOf(_Z^2-3)*x+4*RootOf(_Z^2-3)+6*(x^3+1)^(1/2))/(3*x^2-4*x+
2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {-2+2 x+x^2}{\left (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {9 \, x^{4} - 4 \, \sqrt {3} \sqrt {x^{3} + 1} {\left (3 \, x^{2} - 2 \, x + 4\right )} + 28 \, x^{2} - 16 \, x + 28}{9 \, x^{4} - 24 \, x^{3} + 28 \, x^{2} - 16 \, x + 4}\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*log((9*x^4 - 4*sqrt(3)*sqrt(x^3 + 1)*(3*x^2 - 2*x + 4) + 28*x^2 - 16*x + 28)/(9*x^4 - 24*x^3 + 28*
x^2 - 16*x + 4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.98 (sec) , antiderivative size = 274, normalized size of antiderivative = 8.30 \[ \int \frac {-2+2 x+x^2}{\left (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {5}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {5}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

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

[Out]

-((3^(1/2)*1i + 3)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^
(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((2^(1/2)*1
i)/3 + 5/3), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) - e
llipticF(asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) + ellip
ticPi(-((3^(1/2)*1i)/2 + 3/2)/((2^(1/2)*1i)/3 - 5/3), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)
*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/(3*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^
(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))