3.2.0 ∫ −2−2x+x2 ________________________(2+x2 )√ ____

Integrand size = 25, antiderivative size = 21 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=-2 \text {arctanh}\left (\frac {\sqrt {-1+x^3}}{1+x+x^2}\right ) \]

Rubi [A] (veriﬁed)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2171, 212} \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=2 \text {arctanh}\left (\frac {1-x}{\sqrt {x^3-1}}\right ) \]

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

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])

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-g

/e, Subst[Int[1/(1 + a*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, e, f, g, h}, x] &&

EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*c*g - 4*a*e*h, 0]

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

Mathematica [A] (veriﬁed)

Time = 1.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=-2 \text {arctanh}\left (\frac {\sqrt {-1+x^3}}{1+x+x^2}\right ) \]

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

Maple [A] (veriﬁed)

Time = 2.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33


method	result	size

trager	\(-\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-1}+2 x}{x^{2}+2}\right )\)	\(28\)
default	\(\text {Expression too large to display}\)	\(1656\)
elliptic	\(\text {Expression too large to display}\)	\(1865\)

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

Fricas [A] (veriﬁcation not implemented)

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\log \left (\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{3} - 1}}{x^{2} + 2}\right ) \]

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

log((x^2 + 2*x - 2*sqrt(x^3 - 1))/(x^2 + 2))

Sympy [F]

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

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

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

Maxima [F]

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

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

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

Giac [F]

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

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

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

Mupad [B] (veriﬁcation not implemented)

Time = 0.11 (sec) , antiderivative size = 275, normalized size of antiderivative = 13.10 \[ \int \frac {-2-2 x+x^2}{\left (2+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

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

((3^(1/2)*1i + 3)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((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)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(2^(1/2)*1

i + 1), 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

ticF(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)) + elliptic

Pi(-((3^(1/2)*1i)/2 + 3/2)/(2^(1/2)*1i - 1), 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^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3

\(\int \frac {-2-2 x+x^2}{(2+x^2) \sqrt {-1+x^3}} \, dx\) [192]

Optimal result