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

Optimal. Leaf size=33 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^3+1}}{\sqrt {3} \left (x^2-x+1\right )}\right )}{\sqrt {3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2145, 206} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^3+1}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[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 206

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

Rule 2145

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*} \int \frac {-2+2 x+x^2}{\left (2-4 x+3 x^2\right ) \sqrt {1+x^3}} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{6-2 x^2} \, dx,x,\frac {1+x}{\sqrt {1+x^3}}\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {1+x}{\sqrt {3} \sqrt {1+x^3}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 1.45, size = 394, normalized size = 11.94 \begin {gather*} \frac {2 \sqrt {\frac {x+1}{1+\sqrt [3]{-1}}} \sqrt {x^2-x+1} \left (-\frac {\sqrt {3} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x+1}+\frac {3 i \sqrt {2} \Pi \left (\frac {6 \sqrt {3}}{i-2 \sqrt {2}+3 \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-2 i+3 (-1)^{5/6}+\sqrt {2}}+\frac {15 \Pi \left (\frac {6 \sqrt {3}}{i-2 \sqrt {2}+3 \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) x-2 i}{-3 i+\sqrt {3}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{-2 i+3 (-1)^{5/6}+\sqrt {2}}+\frac {6 i \left (\sqrt {2}+5 i\right ) \Pi \left (\frac {6 \sqrt {3}}{i+2 \sqrt {2}+3 \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) x-2 i}{-3 i+\sqrt {3}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{i+2 \sqrt {2}+3 \sqrt {3}}\right )}{9 \sqrt {x^3+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*(-((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) - x)*EllipticF[Ar
cSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(1 + (-1)^(2/3)*x)) + ((3*I)*Sqrt[2]*EllipticPi[
(6*Sqrt[3])/(I - 2*Sqrt[2] + 3*Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2*I
+ 3*(-1)^(5/6) + Sqrt[2]) + (15*EllipticPi[(6*Sqrt[3])/(I - 2*Sqrt[2] + 3*Sqrt[3]), ArcSin[Sqrt[(-2*I + (I +
Sqrt[3])*x)/(-3*I + Sqrt[3])]], (1 + I*Sqrt[3])/2])/(-2*I + 3*(-1)^(5/6) + Sqrt[2]) + ((6*I)*(5*I + Sqrt[2])*E
llipticPi[(6*Sqrt[3])/(I + 2*Sqrt[2] + 3*Sqrt[3]), ArcSin[Sqrt[(-2*I + (I + Sqrt[3])*x)/(-3*I + Sqrt[3])]], (1
+ I*Sqrt[3])/2])/(I + 2*Sqrt[2] + 3*Sqrt[3])))/(9*Sqrt[1 + x^3])

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IntegrateAlgebraic [A]  time = 1.09, size = 33, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1+x^3}}{\sqrt {3} \left (1-x+x^2\right )}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-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]

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fricas [B]  time = 0.51, size = 66, normalized size = 2.00 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (3 \, x^{2} - 4 \, x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.60, size = 61, normalized size = 1.85

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +4 \RootOf \left (\textit {\_Z}^{2}-3\right )-6 \sqrt {x^{3}+1}}{3 x^{2}-4 x +2}\right )}{3}\)	\(61\)
default	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {4 \left (\frac {5}{6}+\frac {i \sqrt {2}}{6}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}-\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {4 \left (\frac {5}{6}-\frac {i \sqrt {2}}{6}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}+\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\)	\(435\)
elliptic	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}+\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}-\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{9}-\frac {i \sqrt {2}}{9}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{6}+\frac {i \sqrt {2}}{6}-\frac {7 i \sqrt {3}}{18}+\frac {i \left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[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)
)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (3 \, x^{2} - 4 \, x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.21, size = 274, normalized size = 8.30 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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