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

Optimal. Leaf size=231 \[ \frac {\tan ^{-1}\left (\frac {2^{2/3} x}{2 \sqrt {3} \sqrt [3]{x^2-1}+2^{2/3} \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {3\ 2^{2/3} x \sqrt [3]{x^2-1}}{-\sqrt [3]{2} x^2-6 \left (x^2-1\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{x^2-1}-3 \sqrt [3]{2}}\right )}{6\ 2^{2/3}}-\frac {i \tanh ^{-1}\left (\frac {2 i \sqrt [3]{2} \sqrt {3} x-i 2^{2/3} \sqrt {3} x \sqrt [3]{x^2-1}}{\sqrt [3]{2} x^2-6 \left (x^2-1\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{x^2-1}-3 \sqrt [3]{2}}\right )}{6\ 2^{2/3} \sqrt {3}} \]

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Rubi [A]  time = 0.02, antiderivative size = 136, normalized size of antiderivative = 0.59, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {393} \begin {gather*} -\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x^2-1}+1\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{2} \sqrt [3]{x^2-1}+\sqrt [3]{-1}}\right )-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x^2)^(1/3)*(3 + x^2)),x]

[Out]

-1/2*((-1)^(2/3)*ArcTan[Sqrt[3]/x])/(2^(2/3)*Sqrt[3]) - ((-1)^(2/3)*ArcTan[(Sqrt[3]*(1 + (-1)^(2/3)*2^(1/3)*(-
1 + x^2)^(1/3)))/x])/(2*2^(2/3)*Sqrt[3]) + ((-1/2)^(2/3)*ArcTanh[x])/6 - ((-1/2)^(2/3)*ArcTanh[((-1)^(1/3)*x)/
((-1)^(1/3) + 2^(1/3)*(-1 + x^2)^(1/3))])/2

Rule 393

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, Simp[(q*ArcT
an[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (Simp[(q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a +
b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] - Simp[(q*ArcTanh[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[(
Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx &=-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {(-1)^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{-1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{6} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}(x)-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} x}{\sqrt [3]{-1}+\sqrt [3]{2} \sqrt [3]{-1+x^2}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 116, normalized size = 0.50 \begin {gather*} -\frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )}{\sqrt [3]{x^2-1} \left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};x^2,-\frac {x^2}{3}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right )\right )-9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -1/3*x^2])/((-1 + x^2)^(1/3)*(3 + x^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, x^
2, -1/3*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -1/3*x^2] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -1/3*x^2])))

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IntegrateAlgebraic [F]  time = 3.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{-1+x^2} \left (3+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 2.50, size = 1896, normalized size = 8.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20736*432^(5/6)*sqrt(3)*log(10368*(6*2^(2/3)*(x^6 + 225*x^4 - 189*x^2 + 27) + 144*432^(1/6)*sqrt(3)*(x^5 - x
^3) + (432^(5/6)*sqrt(3)*(7*x^3 - 3*x) + 216*2^(1/3)*(x^4 + 3*x^2))*(x^2 - 1)^(2/3) + 72*(x^5 + 18*x^4 + 24*x^
3 - 18*x^2 - 9*x)*(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1/20736*432^(5/6)*sqrt(3)*log(2592*(6*2^(2/3
)*(x^6 + 225*x^4 - 189*x^2 + 27) + 144*432^(1/6)*sqrt(3)*(x^5 - x^3) + (432^(5/6)*sqrt(3)*(7*x^3 - 3*x) + 216*
2^(1/3)*(x^4 + 3*x^2))*(x^2 - 1)^(2/3) + 72*(x^5 + 18*x^4 + 24*x^3 - 18*x^2 - 9*x)*(x^2 - 1)^(1/3))/(x^6 + 9*x
^4 + 27*x^2 + 27)) - 1/20736*432^(5/6)*sqrt(3)*log(10368*(6*2^(2/3)*(x^6 + 225*x^4 - 189*x^2 + 27) - 144*432^(
1/6)*sqrt(3)*(x^5 - x^3) - (432^(5/6)*sqrt(3)*(7*x^3 - 3*x) - 216*2^(1/3)*(x^4 + 3*x^2))*(x^2 - 1)^(2/3) - 72*
(x^5 - 18*x^4 + 24*x^3 + 18*x^2 - 9*x)*(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) - 1/20736*432^(5/6)*sqrt(
3)*log(2592*(6*2^(2/3)*(x^6 + 225*x^4 - 189*x^2 + 27) - 144*432^(1/6)*sqrt(3)*(x^5 - x^3) - (432^(5/6)*sqrt(3)
*(7*x^3 - 3*x) - 216*2^(1/3)*(x^4 + 3*x^2))*(x^2 - 1)^(2/3) - 72*(x^5 - 18*x^4 + 24*x^3 + 18*x^2 - 9*x)*(x^2 -
1)^(1/3))/(x^6 + 9*x^4 + 27*x^2 + 27)) + 1/1296*432^(5/6)*arctan(-1/36*(432^(5/6)*(x^5 - 18*x^3 + 9*x)*(x^2 -
1)^(1/3) - sqrt(3)*2^(1/3)*(432^(5/6)*(x^4 + 9*x^2)*(x^2 - 1)^(2/3) + 288*sqrt(3)*(2*x^4 - 3*x^2)*(x^2 - 1)^(
1/3) + 6*432^(1/6)*(x^6 + 141*x^4 - 153*x^2 + 27)) + 648*432^(1/6)*(3*x^3 - x)*(x^2 - 1)^(2/3) + 72*sqrt(3)*(7
*x^5 + 6*x^3 - 9*x))/(x^6 - 225*x^4 + 243*x^2 - 27)) + 1/2592*432^(5/6)*arctan(-1/18*(sqrt(2)*(18*sqrt(3)*2^(2
/3)*(29*x^11 + 879*x^9 - 12078*x^7 + 10638*x^5 - 3807*x^3 + 243*x) - 2*(x^2 - 1)^(2/3)*(432^(5/6)*(x^10 + 153*
x^8 - 1701*x^6 + 459*x^4) - 216*sqrt(3)*2^(1/3)*(31*x^9 - 297*x^7 - 27*x^5 - 27*x^3)) + 36*(x^2 - 1)^(1/3)*(sq
rt(3)*(x^11 + 1167*x^9 - 13158*x^7 + 17550*x^5 - 4779*x^3 + 243*x) - 8*sqrt(3)*(13*x^10 - 6*x^8 - 1404*x^6 + 1
350*x^4 - 81*x^2)) - 3*432^(1/6)*(x^12 + 7620*x^10 - 92115*x^8 + 169776*x^6 - 109269*x^4 + 16524*x^2 - 729))*s
qrt((6*2^(2/3)*(x^6 + 225*x^4 - 189*x^2 + 27) + 144*432^(1/6)*sqrt(3)*(x^5 - x^3) + (432^(5/6)*sqrt(3)*(7*x^3
- 3*x) + 216*2^(1/3)*(x^4 + 3*x^2))*(x^2 - 1)^(2/3) + 72*(x^5 + 18*x^4 + 24*x^3 - 18*x^2 - 9*x)*(x^2 - 1)^(1/3
))/(x^6 + 9*x^4 + 27*x^2 + 27)) - 216*(sqrt(3)*2^(2/3)*(x^10 + 144*x^8 - 918*x^6 + 2808*x^4 - 243*x^2) - 3*432
^(1/6)*(31*x^9 - 568*x^7 + 1710*x^5 - 432*x^3 + 27*x))*(x^2 - 1)^(2/3) - 18*sqrt(3)*(x^12 - 366*x^10 + 14535*x
^8 - 42660*x^6 + 58239*x^4 - 14094*x^2 + 729) + 144*sqrt(3)*(11*x^11 - 807*x^9 + 4518*x^7 - 5238*x^5 + 3807*x^
3 - 243*x) + (x^2 - 1)^(1/3)*(432^(5/6)*(x^11 - 1215*x^9 + 11754*x^7 - 21006*x^5 + 5589*x^3 - 243*x) - 432*sqr
t(3)*2^(1/3)*(13*x^10 - 120*x^8 + 1242*x^6 - 1728*x^4 + 81*x^2)))/(x^12 - 8334*x^10 + 110727*x^8 - 301860*x^6
+ 187839*x^4 - 21870*x^2 + 729)) + 1/2592*432^(5/6)*arctan(1/18*(sqrt(2)*(18*sqrt(3)*2^(2/3)*(29*x^11 + 879*x^
9 - 12078*x^7 + 10638*x^5 - 3807*x^3 + 243*x) + 2*(x^2 - 1)^(2/3)*(432^(5/6)*(x^10 + 153*x^8 - 1701*x^6 + 459*
x^4) + 216*sqrt(3)*2^(1/3)*(31*x^9 - 297*x^7 - 27*x^5 - 27*x^3)) + 36*(x^2 - 1)^(1/3)*(sqrt(3)*(x^11 + 1167*x^
9 - 13158*x^7 + 17550*x^5 - 4779*x^3 + 243*x) + 8*sqrt(3)*(13*x^10 - 6*x^8 - 1404*x^6 + 1350*x^4 - 81*x^2)) +
3*432^(1/6)*(x^12 + 7620*x^10 - 92115*x^8 + 169776*x^6 - 109269*x^4 + 16524*x^2 - 729))*sqrt((6*2^(2/3)*(x^6 +
225*x^4 - 189*x^2 + 27) - 144*432^(1/6)*sqrt(3)*(x^5 - x^3) - (432^(5/6)*sqrt(3)*(7*x^3 - 3*x) - 216*2^(1/3)*
(x^4 + 3*x^2))*(x^2 - 1)^(2/3) - 72*(x^5 - 18*x^4 + 24*x^3 + 18*x^2 - 9*x)*(x^2 - 1)^(1/3))/(x^6 + 9*x^4 + 27*
x^2 + 27)) - 216*(sqrt(3)*2^(2/3)*(x^10 + 144*x^8 - 918*x^6 + 2808*x^4 - 243*x^2) + 3*432^(1/6)*(31*x^9 - 568*
x^7 + 1710*x^5 - 432*x^3 + 27*x))*(x^2 - 1)^(2/3) - 18*sqrt(3)*(x^12 - 366*x^10 + 14535*x^8 - 42660*x^6 + 5823
9*x^4 - 14094*x^2 + 729) - 144*sqrt(3)*(11*x^11 - 807*x^9 + 4518*x^7 - 5238*x^5 + 3807*x^3 - 243*x) - (x^2 - 1
)^(1/3)*(432^(5/6)*(x^11 - 1215*x^9 + 11754*x^7 - 21006*x^5 + 5589*x^3 - 243*x) + 432*sqrt(3)*2^(1/3)*(13*x^10
- 120*x^8 + 1242*x^6 - 1728*x^4 + 81*x^2)))/(x^12 - 8334*x^10 + 110727*x^8 - 301860*x^6 + 187839*x^4 - 21870*
x^2 + 729))

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

[Out]

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

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maple [C]  time = 20.14, size = 874, normalized size = 3.78

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{6}+108\right ) \ln \left (\frac {9072 x^{3} \left (x^{2}-1\right )^{\frac {2}{3}}-3888 x \left (x^{2}-1\right )^{\frac {2}{3}}-486 \RootOf \left (\textit {\_Z}^{6}+108\right )-27 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}+72 x^{3} \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}+189 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{2}-225 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{4}-72 x^{5} \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}-1296 x^{5} \RootOf \left (\textit {\_Z}^{6}+108\right )-\RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{6}+1296 \RootOf \left (\textit {\_Z}^{6}+108\right ) x^{3}-4050 \RootOf \left (\textit {\_Z}^{6}+108\right ) x^{4}+3402 \RootOf \left (\textit {\_Z}^{6}+108\right ) x^{2}-18 \RootOf \left (\textit {\_Z}^{6}+108\right ) x^{6}+1296 \left (x^{2}-1\right )^{\frac {2}{3}} x^{4}+3888 \left (x^{2}-1\right )^{\frac {2}{3}} x^{2}+6 \RootOf \left (\textit {\_Z}^{6}+108\right )^{5} \left (x^{2}-1\right )^{\frac {1}{3}} x^{5}+108 \RootOf \left (\textit {\_Z}^{6}+108\right )^{5} \left (x^{2}-1\right )^{\frac {1}{3}} x^{4}+144 \RootOf \left (\textit {\_Z}^{6}+108\right )^{5} \left (x^{2}-1\right )^{\frac {1}{3}} x^{3}+36 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{5}+648 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{4}+864 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{3}-648 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-324 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x -108 \RootOf \left (\textit {\_Z}^{6}+108\right )^{5} \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-54 \RootOf \left (\textit {\_Z}^{6}+108\right )^{5} \left (x^{2}-1\right )^{\frac {1}{3}} x}{\left (x^{2}+3\right )^{3}}\right )}{36}+\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{6}+72 x^{5} \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}+225 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{4}-36 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{5}-72 x^{3} \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}-648 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{4}-189 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{2}-864 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{3}+648 \left (x^{2}-1\right )^{\frac {2}{3}} x^{4}+648 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+4536 x^{3} \left (x^{2}-1\right )^{\frac {2}{3}}+27 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}+324 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x +1944 \left (x^{2}-1\right )^{\frac {2}{3}} x^{2}-1944 x \left (x^{2}-1\right )^{\frac {2}{3}}}{\left (x^{2}+3\right )^{3}}\right ) \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}}{432}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{6}+72 x^{5} \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}+225 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{4}-36 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{5}-72 x^{3} \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}-648 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{4}-189 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4} x^{2}-864 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{3}+648 \left (x^{2}-1\right )^{\frac {2}{3}} x^{4}+648 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+4536 x^{3} \left (x^{2}-1\right )^{\frac {2}{3}}+27 \RootOf \left (\textit {\_Z}^{6}+108\right )^{4}+324 \RootOf \left (\textit {\_Z}^{6}+108\right )^{2} \left (x^{2}-1\right )^{\frac {1}{3}} x +1944 \left (x^{2}-1\right )^{\frac {2}{3}} x^{2}-1944 x \left (x^{2}-1\right )^{\frac {2}{3}}}{\left (x^{2}+3\right )^{3}}\right ) \RootOf \left (\textit {\_Z}^{6}+108\right )}{72}\)	\(874\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*RootOf(_Z^6+108)*ln((9072*x^3*(x^2-1)^(2/3)-3888*x*(x^2-1)^(2/3)+72*x^3*RootOf(_Z^6+108)^4+189*RootOf(_Z
^6+108)^4*x^2-225*RootOf(_Z^6+108)^4*x^4-72*x^5*RootOf(_Z^6+108)^4-1296*x^5*RootOf(_Z^6+108)-RootOf(_Z^6+108)^
4*x^6+1296*RootOf(_Z^6+108)*x^3-4050*RootOf(_Z^6+108)*x^4+3402*RootOf(_Z^6+108)*x^2-27*RootOf(_Z^6+108)^4-486*
RootOf(_Z^6+108)-18*RootOf(_Z^6+108)*x^6+1296*(x^2-1)^(2/3)*x^4+3888*(x^2-1)^(2/3)*x^2+6*RootOf(_Z^6+108)^5*(x
^2-1)^(1/3)*x^5+108*RootOf(_Z^6+108)^5*(x^2-1)^(1/3)*x^4+144*RootOf(_Z^6+108)^5*(x^2-1)^(1/3)*x^3+36*RootOf(_Z
^6+108)^2*(x^2-1)^(1/3)*x^5+648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^4+864*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^3-
648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^2-324*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x-108*RootOf(_Z^6+108)^5*(x^2-1)
^(1/3)*x^2-54*RootOf(_Z^6+108)^5*(x^2-1)^(1/3)*x)/(x^2+3)^3)+1/432*ln((RootOf(_Z^6+108)^4*x^6+72*x^5*RootOf(_Z
^6+108)^4+225*RootOf(_Z^6+108)^4*x^4-36*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^5-72*x^3*RootOf(_Z^6+108)^4-648*Roo
tOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^4-189*RootOf(_Z^6+108)^4*x^2-864*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^3+648*(x^2
-1)^(2/3)*x^4+648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^2+4536*x^3*(x^2-1)^(2/3)+27*RootOf(_Z^6+108)^4+324*RootOf
(_Z^6+108)^2*(x^2-1)^(1/3)*x+1944*(x^2-1)^(2/3)*x^2-1944*x*(x^2-1)^(2/3))/(x^2+3)^3)*RootOf(_Z^6+108)^4-1/72*l
n((RootOf(_Z^6+108)^4*x^6+72*x^5*RootOf(_Z^6+108)^4+225*RootOf(_Z^6+108)^4*x^4-36*RootOf(_Z^6+108)^2*(x^2-1)^(
1/3)*x^5-72*x^3*RootOf(_Z^6+108)^4-648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^4-189*RootOf(_Z^6+108)^4*x^2-864*Roo
tOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^3+648*(x^2-1)^(2/3)*x^4+648*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x^2+4536*x^3*(x^2
-1)^(2/3)+27*RootOf(_Z^6+108)^4+324*RootOf(_Z^6+108)^2*(x^2-1)^(1/3)*x+1944*(x^2-1)^(2/3)*x^2-1944*x*(x^2-1)^(
2/3))/(x^2+3)^3)*RootOf(_Z^6+108)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 - 1)^(1/3)*(x^2 + 3)),x)

[Out]

int(1/((x^2 - 1)^(1/3)*(x^2 + 3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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