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

Optimal. Leaf size=116 \[ \frac {3 \left (x^2-1\right )^{2/3}}{x+1}-\frac {7}{4} \log \left (2 \sqrt [3]{x^2-1}+x-1\right )+\frac {7}{8} \log \left (x^2+4 \left (x^2-1\right )^{2/3}+(2-2 x) \sqrt [3]{x^2-1}-2 x+1\right )-\frac {7}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2-1}}{\sqrt [3]{x^2-1}-x+1}\right ) \]

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*x)/(1 + Sqrt[3] + (-1 + x^2)^(1/3)) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + (-1 + x^2)^(1/3))*Sqrt[(1 - (-1 + x
^2)^(1/3) + (-1 + x^2)^(2/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] + (-1 + x^2)^(
1/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*x*Sqrt[(1 + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1
+ x^2)^(1/3))^2]) + (Sqrt[2]*3^(3/4)*(1 + (-1 + x^2)^(1/3))*Sqrt[(1 - (-1 + x^2)^(1/3) + (-1 + x^2)^(2/3))/(1
+ Sqrt[3] + (-1 + x^2)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1 + x^2)^(
1/3))], -7 - 4*Sqrt[3]])/(x*Sqrt[(1 + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))^2]) - Defer[Int][(1 +
5*x)/((-1 + x^2)^(1/3)*(3 + 4*x + x^2)), x]

Rubi steps

\begin {align*} \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{-1+x^2}}-\frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ &=\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ &=\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}+\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{x}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ &=\frac {3 x}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{2 x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.17, size = 71, normalized size = 0.61 \begin {gather*} \frac {3 \left (x^2-1\right )^{2/3}}{x+1}-\frac {21 \left (x^2-1\right )^{2/3} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};\frac {1-x}{2},\frac {1-x}{4}\right )}{8 \sqrt [3]{2} (x+1)^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*(-1 + x^2)^(2/3))/(1 + x) - (21*(-1 + x^2)^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, (1 - x)/2, (1 - x)/4])/(8*2^(1/
3)*(1 + x)^(2/3))

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IntegrateAlgebraic [A]  time = 0.36, size = 116, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^2\right )^{2/3}}{1+x}-\frac {7}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{1-x+\sqrt [3]{-1+x^2}}\right )-\frac {7}{4} \log \left (-1+x+2 \sqrt [3]{-1+x^2}\right )+\frac {7}{8} \log \left (1-2 x+x^2+(2-2 x) \sqrt [3]{-1+x^2}+4 \left (-1+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^2)^(2/3))/(1 + x) - (7*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(1 - x + (-1 + x^2)^(1/3))])/4 - (
7*Log[-1 + x + 2*(-1 + x^2)^(1/3)])/4 + (7*Log[1 - 2*x + x^2 + (2 - 2*x)*(-1 + x^2)^(1/3) + 4*(-1 + x^2)^(2/3)
])/8

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fricas [A]  time = 0.88, size = 127, normalized size = 1.09 \begin {gather*} \frac {14 \, \sqrt {3} {\left (x + 1\right )} \arctan \left (\frac {286273 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (66978 \, x^{2} + 434719 \, x + 635653\right )} + 539695 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{226981 \, x^{2} - 1974837 \, x - 1293894}\right ) - 7 \, {\left (x + 1\right )} \log \left (\frac {x^{2} + 6 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 6 \, x + 12 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 9}{x^{2} + 6 \, x + 9}\right ) + 24 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(14*sqrt(3)*(x + 1)*arctan((286273*sqrt(3)*(x^2 - 1)^(1/3)*(x - 1) + sqrt(3)*(66978*x^2 + 434719*x + 63565
3) + 539695*sqrt(3)*(x^2 - 1)^(2/3))/(226981*x^2 - 1974837*x - 1293894)) - 7*(x + 1)*log((x^2 + 6*(x^2 - 1)^(1
/3)*(x - 1) + 6*x + 12*(x^2 - 1)^(2/3) + 9)/(x^2 + 6*x + 9)) + 24*(x^2 - 1)^(2/3))/(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.65, size = 398, normalized size = 3.43

method	result	size
trager	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{1+x}-\frac {7 \ln \left (-\frac {4032 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-12096 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +2592 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-1548 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -1404 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+1548 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+9000 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -948 \left (x^{2}-1\right )^{\frac {2}{3}}-216 x \left (x^{2}-1\right )^{\frac {1}{3}}+119 x^{2}+4788 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+216 \left (x^{2}-1\right )^{\frac {1}{3}}-1326 x -969}{\left (3+x \right )^{2}}\right )}{4}+\frac {21 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {2448 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-7344 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +1296 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+1422 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -387 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-1422 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-1746 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x +258 \left (x^{2}-1\right )^{\frac {2}{3}}-108 x \left (x^{2}-1\right )^{\frac {1}{3}}+7 x^{2}-2907 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+108 \left (x^{2}-1\right )^{\frac {1}{3}}+378 x +399}{\left (3+x \right )^{2}}\right )}{2}\)	\(398\)
risch	\(\frac {-3+3 x}{\left (x^{2}-1\right )^{\frac {1}{3}}}-\frac {7 \ln \left (\frac {-448 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+864 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-516 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +1344 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -20 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+516 \left (x^{2}-1\right )^{\frac {2}{3}}+516 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+474 x \left (x^{2}-1\right )^{\frac {1}{3}}+1656 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x +3 x^{2}-474 \left (x^{2}-1\right )^{\frac {1}{3}}+1596 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+162 x +171}{\left (3+x \right )^{2}}\right )}{4}+\frac {7 \ln \left (-\frac {-544 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+864 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+948 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +1632 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +286 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-948 \left (x^{2}-1\right )^{\frac {2}{3}}-948 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-258 x \left (x^{2}-1\right )^{\frac {1}{3}}-2796 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -21 x^{2}+258 \left (x^{2}-1\right )^{\frac {1}{3}}-1938 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+234 x +171}{\left (3+x \right )^{2}}\right )}{4}-\frac {7 \ln \left (-\frac {-544 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+864 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+948 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +1632 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +286 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-948 \left (x^{2}-1\right )^{\frac {2}{3}}-948 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-258 x \left (x^{2}-1\right )^{\frac {1}{3}}-2796 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -21 x^{2}+258 \left (x^{2}-1\right )^{\frac {1}{3}}-1938 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+234 x +171}{\left (3+x \right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{2}\)	\(582\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x^2-1)^(2/3)/(1+x)-7/4*ln(-(4032*RootOf(36*_Z^2-6*_Z+1)^2*x^2-12096*RootOf(36*_Z^2-6*_Z+1)^2*x+2592*RootOf(
36*_Z^2-6*_Z+1)*(x^2-1)^(2/3)-1548*RootOf(36*_Z^2-6*_Z+1)*(x^2-1)^(1/3)*x-1404*RootOf(36*_Z^2-6*_Z+1)*x^2+1548
*RootOf(36*_Z^2-6*_Z+1)*(x^2-1)^(1/3)+9000*RootOf(36*_Z^2-6*_Z+1)*x-948*(x^2-1)^(2/3)-216*x*(x^2-1)^(1/3)+119*
x^2+4788*RootOf(36*_Z^2-6*_Z+1)+216*(x^2-1)^(1/3)-1326*x-969)/(3+x)^2)+21/2*RootOf(36*_Z^2-6*_Z+1)*ln((2448*Ro
otOf(36*_Z^2-6*_Z+1)^2*x^2-7344*RootOf(36*_Z^2-6*_Z+1)^2*x+1296*RootOf(36*_Z^2-6*_Z+1)*(x^2-1)^(2/3)+1422*Root
Of(36*_Z^2-6*_Z+1)*(x^2-1)^(1/3)*x-387*RootOf(36*_Z^2-6*_Z+1)*x^2-1422*RootOf(36*_Z^2-6*_Z+1)*(x^2-1)^(1/3)-17
46*RootOf(36*_Z^2-6*_Z+1)*x+258*(x^2-1)^(2/3)-108*x*(x^2-1)^(1/3)+7*x^2-2907*RootOf(36*_Z^2-6*_Z+1)+108*(x^2-1
)^(1/3)+378*x+399)/(3+x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2-x+2}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+4\,x+3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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