3.6.81 \(\int \frac {1}{(\sqrt {3}-x) \sqrt [3]{1+x^2}} \, dx\)
Optimal. Leaf size=101 \[ -\frac {\log \left (-\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+1}+x+\sqrt {3}\right )}{2\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \left (x+\sqrt {3}\right )}{3 \sqrt [3]{x^2+1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (\sqrt {3}-x\right )}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used =
{751} \begin {gather*} -\frac {\log \left (-\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+1}+x+\sqrt {3}\right )}{2\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \left (x+\sqrt {3}\right )}{3 \sqrt [3]{x^2+1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (\sqrt {3}-x\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((Sqrt[3] - x)*(1 + x^2)^(1/3)),x]
[Out]
ArcTan[1/Sqrt[3] + (2^(2/3)*(Sqrt[3] + x))/(3*(1 + x^2)^(1/3))]/(2^(2/3)*Sqrt[3]) + Log[Sqrt[3] - x]/(2*2^(2/3
)) - Log[Sqrt[3] + x - 2^(1/3)*Sqrt[3]*(1 + x^2)^(1/3)]/(2*2^(2/3))
Rule 751
Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp
[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c
*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]
Rubi steps
\begin {align*} \int \frac {1}{\left (\sqrt {3}-x\right ) \sqrt [3]{1+x^2}} \, dx &=\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \left (\sqrt {3}+x\right )}{3 \sqrt [3]{1+x^2}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (\sqrt {3}-x\right )}{2\ 2^{2/3}}-\frac {\log \left (\sqrt {3}+x-\sqrt [3]{2} \sqrt {3} \sqrt [3]{1+x^2}\right )}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 110, normalized size = 1.09 \begin {gather*} \frac {3 \sqrt [3]{\frac {x-i}{x-\sqrt {3}}} \sqrt [3]{\frac {x+i}{x-\sqrt {3}}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {-i+\sqrt {3}}{\sqrt {3}-x},\frac {i+\sqrt {3}}{\sqrt {3}-x}\right )}{2 \sqrt [3]{x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((Sqrt[3] - x)*(1 + x^2)^(1/3)),x]
[Out]
(3*((-I + x)/(-Sqrt[3] + x))^(1/3)*((I + x)/(-Sqrt[3] + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (-I + Sqrt[3])/
(Sqrt[3] - x), (I + Sqrt[3])/(Sqrt[3] - x)])/(2*(1 + x^2)^(1/3))
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IntegrateAlgebraic [A] time = 0.28, size = 189, normalized size = 1.87 \begin {gather*} -\frac {\log \left (-6 \sqrt [3]{x^2+1}+2^{2/3} \sqrt {3} x+3\ 2^{2/3}\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2} x^2+2^{2/3} \sqrt {3} \sqrt [3]{x^2+1} x+6 \left (x^2+1\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{x^2+1}+2 \sqrt [3]{2} \sqrt {3} x+3 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {1}{3} 2^{2/3} x+\frac {2^{2/3}}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right )}{2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((Sqrt[3] - x)*(1 + x^2)^(1/3)),x]
[Out]
ArcTan[(2^(2/3)/Sqrt[3] + (2^(2/3)*x)/3 + (1 + x^2)^(1/3)/Sqrt[3])/(1 + x^2)^(1/3)]/(2^(2/3)*Sqrt[3]) - Log[3*
2^(2/3) + 2^(2/3)*Sqrt[3]*x - 6*(1 + x^2)^(1/3)]/(3*2^(2/3)) + Log[3*2^(1/3) + 2*2^(1/3)*Sqrt[3]*x + 2^(1/3)*x
^2 + 3*2^(2/3)*(1 + x^2)^(1/3) + 2^(2/3)*Sqrt[3]*x*(1 + x^2)^(1/3) + 6*(1 + x^2)^(2/3)]/(6*2^(2/3))
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fricas [B] time = 3.39, size = 335, normalized size = 3.32 \begin {gather*} \frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} {\left (8 \, \sqrt {3} \left (-1\right )^{\frac {2}{3}} x^{3} - \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 18 \, x^{2} - 27\right )}\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{6} + 99 \, x^{4} + 243 \, x^{2} - 12 \, \sqrt {3} {\left (x^{5} + 10 \, x^{3} + 9 \, x\right )} + 81\right )} - 4 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - 42 \, x^{3} - 27 \, x\right )} - 3 \, \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + 18 \, x^{2} + 27\right )}\right )}\right )}}{6 \, {\left (x^{6} - 225 \, x^{4} - 405 \, x^{2} - 243\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 1\right )}^{\frac {2}{3}} {\left (2 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} x + \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )}\right )} - 4^{\frac {1}{3}} {\left (4 \, \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 3 \, x\right )} + \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 18 \, x^{2} + 9\right )}\right )} - 2 \, {\left (9 \, x^{2} + \sqrt {3} {\left (x^{3} + 9 \, x\right )} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} - 6 \, x^{2} + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {1}{3}} {\left (x^{2} + 2 \, \sqrt {3} x + 3\right )} + 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {3} \left (-1\right )^{\frac {1}{3}} x + 3 \, \left (-1\right )^{\frac {1}{3}}\right )}}{x^{2} - 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^2+1)^(1/3)/(-x+3^(1/2)),x, algorithm="fricas")
[Out]
1/6*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(8*sqrt(3)*(-1)^(2/3)*x^3 - (-1)^(2/3)*(x
^4 - 18*x^2 - 27))*(x^2 + 1)^(2/3) - 4^(1/3)*(x^6 + 99*x^4 + 243*x^2 - 12*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 81) -
4*(x^2 + 1)^(1/3)*(sqrt(3)*(-1)^(1/3)*(x^5 - 42*x^3 - 27*x) - 3*(-1)^(1/3)*(7*x^4 + 18*x^2 + 27)))/(x^6 - 225
*x^4 - 405*x^2 - 243)) - 1/24*4^(2/3)*(-1)^(1/3)*log(-(3*4^(2/3)*(x^2 + 1)^(2/3)*(2*sqrt(3)*(-1)^(1/3)*x + (-1
)^(1/3)*(x^2 + 3)) - 4^(1/3)*(4*sqrt(3)*(-1)^(2/3)*(x^3 + 3*x) + (-1)^(2/3)*(x^4 + 18*x^2 + 9)) - 2*(9*x^2 + s
qrt(3)*(x^3 + 9*x) + 9)*(x^2 + 1)^(1/3))/(x^4 - 6*x^2 + 9)) + 1/12*4^(2/3)*(-1)^(1/3)*log(-(4^(1/3)*(x^2 + 2*s
qrt(3)*x + 3) + 2*(x^2 + 1)^(1/3)*(sqrt(3)*(-1)^(1/3)*x + 3*(-1)^(1/3)))/(x^2 - 3))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - \sqrt {3}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^2+1)^(1/3)/(-x+3^(1/2)),x, algorithm="giac")
[Out]
integrate(-1/((x^2 + 1)^(1/3)*(x - sqrt(3))), x)
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maple [C] time = 41.22, size = 2443, normalized size = 24.19 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2+1)^(1/3)/(-x+3^(1/2)),x)
[Out]
1/18*3^(1/2)*(ln((144*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2)
)*_Z+36*_Z^2)*(x^2+1)^(2/3)*x-2*RootOf(_Z^3-6*3^(1/2))^3*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/
2))*_Z+36*_Z^2)*x^3-60*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36
*_Z^2)^2*x^3+54*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x+864*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/
2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x+9*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x^
2+144*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2
+1)^(1/3)*x^2+162*(x^2+1)^(2/3)*x+3^(1/2)*RootOf(_Z^3-6*3^(1/2))*x^3+30*3^(1/2)*RootOf(RootOf(_Z^3-6*3^(1/2))^
2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3-18*RootOf(_Z^3-6*3^(1/2))^3*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootO
f(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x-540*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3
^(1/2))*_Z+36*_Z^2)^2*x+432*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*
_Z+36*_Z^2)*(x^2+1)^(2/3)+27*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)+432*3^(1/2)*RootOf(_Z^3-6*3^(1/2))
*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)+9*RootOf(_Z^3-6*3^(1/2))*3
^(1/2)*x+270*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*x+162*3^(1/2)*(x^2+1
)^(2/3)+63*x^2*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2
)*x^2+63*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2))/(3^
(1/2)*x-3)^3)*RootOf(_Z^3-6*3^(1/2))+6*ln((144*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^
2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(2/3)*x-2*RootOf(_Z^3-6*3^(1/2))^3*RootOf(RootOf(_Z^3-6*3^(1/2)
)^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3-60*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*Roo
tOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*x^3+54*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x+864*RootOf(_Z^3-6*3^(1/2))*R
ootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x+9*3^(1/2)*RootOf(_Z^3-6*3^
(1/2))^2*(x^2+1)^(1/3)*x^2+144*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*
3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x^2+162*(x^2+1)^(2/3)*x+3^(1/2)*RootOf(_Z^3-6*3^(1/2))*x^3+30*3^(1/2)*RootO
f(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3-18*RootOf(_Z^3-6*3^(1/2))^3*RootOf(RootOf(
_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x-540*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(
1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*x+432*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+
6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(2/3)+27*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)+432*3^(1/
2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)+9
*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x+270*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(
1/2)*x+162*3^(1/2)*(x^2+1)^(2/3)+63*x^2*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_
Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^2+63*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*
3^(1/2))*_Z+36*_Z^2))/(3^(1/2)*x-3)^3)*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)-6*
RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*ln(-(144*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2
*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(2/3)*x+8*RootOf(_Z^3-6*3^(1/2))
^3*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3+60*RootOf(_Z^3-6*3^(1/2))^2*RootOf
(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*x^3+90*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)
*x+864*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/
3)*x+15*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x^2+144*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z
^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x^2+270*(x^2+1)^(2/3)*x+12*3^(1/2)*RootOf(_
Z^3-6*3^(1/2))*x^3+90*3^(1/2)*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3+72*Root
Of(_Z^3-6*3^(1/2))^3*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x+540*RootOf(_Z^3-6*
3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*x+432*RootOf(_Z^3-6*3^(1/2))
^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(2/3)+45*3^(1/2)*RootOf(_Z^3-6
*3^(1/2))^2*(x^2+1)^(1/3)+432*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3
^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)+108*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x+810*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*R
ootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*x+270*3^(1/2)*(x^2+1)^(2/3)+252*x^2*RootOf(_Z^3-6*3^(1/2))+1890*Root
Of(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^2+252*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(Ro
otOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2))/(3^(1/2)*x-3)^3))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - \sqrt {3}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^2+1)^(1/3)/(-x+3^(1/2)),x, algorithm="maxima")
[Out]
-integrate(1/((x^2 + 1)^(1/3)*(x - sqrt(3))), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{{\left (x^2+1\right )}^{1/3}\,\left (x-\sqrt {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 - 3^(1/2))),x)
[Out]
-int(1/((x^2 + 1)^(1/3)*(x - 3^(1/2))), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x \sqrt [3]{x^{2} + 1} - \sqrt {3} \sqrt [3]{x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x**2+1)**(1/3)/(-x+3**(1/2)),x)
[Out]
-Integral(1/(x*(x**2 + 1)**(1/3) - sqrt(3)*(x**2 + 1)**(1/3)), x)
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