3.19.83 \(\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx\)
Optimal. Leaf size=175 \[ \frac {\log \left (-3 \sqrt [3]{x^2+3 x+3}+\sqrt [3]{3} x+3 \sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (3^{2/3} x^2+9 \left (x^2+3 x+3\right )^{2/3}+\left (3 \sqrt [3]{3} x+9 \sqrt [3]{3}\right ) \sqrt [3]{x^2+3 x+3}+6\ 3^{2/3} x+9\ 3^{2/3}\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+3 x+3}}{\sqrt {3}}+\frac {2 x}{3 \sqrt [6]{3}}+\frac {2}{\sqrt [6]{3}}}{\sqrt [3]{x^2+3 x+3}}\right )}{3^{5/6}} \]
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Rubi [A] time = 0.02, antiderivative size = 83, normalized size of antiderivative = 0.47,
number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used =
{752} \begin {gather*} \frac {\log \left (3^{2/3} \sqrt [3]{x^2+3 x+3}-x-3\right )}{2 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {2 (x+3)}{3 \sqrt [6]{3} \sqrt [3]{x^2+3 x+3}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log (x)}{2 \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x*(3 + 3*x + x^2)^(1/3)),x]
[Out]
-(ArcTan[1/Sqrt[3] + (2*(3 + x))/(3*3^(1/6)*(3 + 3*x + x^2)^(1/3))]/3^(5/6)) - Log[x]/(2*3^(1/3)) + Log[-3 - x
+ 3^(2/3)*(3 + 3*x + x^2)^(1/3)]/(2*3^(1/3))
Rule 752
Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[-3*c*e^2*(2*c
*d - b*e), 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] - (2*(c*d - b*e - c*e*x))/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/
3))])/q^2, x] + (-Simp[(3*c*e*Log[d + e*x])/(2*q^2), x] + Simp[(3*c*e*Log[c*d - b*e - c*e*x + q*(a + b*x + c*x
^2)^(1/3)])/(2*q^2), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e
^2 - 3*a*c*e^2, 0] && NegQ[c*e^2*(2*c*d - b*e)]
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 (3+x)}{3 \sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}\right )}{3^{5/6}}-\frac {\log (x)}{2 \sqrt [3]{3}}+\frac {\log \left (-3-x+3^{2/3} \sqrt [3]{3+3 x+x^2}\right )}{2 \sqrt [3]{3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 114, normalized size = 0.65 \begin {gather*} -\frac {3 \sqrt [3]{\frac {2 x-i \sqrt {3}+3}{x}} \sqrt [3]{\frac {2 x+i \sqrt {3}+3}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {3+i \sqrt {3}}{2 x},\frac {i \left (3 i+\sqrt {3}\right )}{2 x}\right )}{2\ 2^{2/3} \sqrt [3]{x^2+3 x+3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(x*(3 + 3*x + x^2)^(1/3)),x]
[Out]
(-3*((3 - I*Sqrt[3] + 2*x)/x)^(1/3)*((3 + I*Sqrt[3] + 2*x)/x)^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -1/2*(3 + I*S
qrt[3])/x, ((I/2)*(3*I + Sqrt[3]))/x])/(2*2^(2/3)*(3 + 3*x + x^2)^(1/3))
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IntegrateAlgebraic [A] time = 0.24, size = 175, normalized size = 1.00 \begin {gather*} \frac {\log \left (-3 \sqrt [3]{x^2+3 x+3}+\sqrt [3]{3} x+3 \sqrt [3]{3}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (3^{2/3} x^2+9 \left (x^2+3 x+3\right )^{2/3}+\left (3 \sqrt [3]{3} x+9 \sqrt [3]{3}\right ) \sqrt [3]{x^2+3 x+3}+6\ 3^{2/3} x+9\ 3^{2/3}\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+3 x+3}}{\sqrt {3}}+\frac {2 x}{3 \sqrt [6]{3}}+\frac {2}{\sqrt [6]{3}}}{\sqrt [3]{x^2+3 x+3}}\right )}{3^{5/6}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x*(3 + 3*x + x^2)^(1/3)),x]
[Out]
-(ArcTan[(2/3^(1/6) + (2*x)/(3*3^(1/6)) + (3 + 3*x + x^2)^(1/3)/Sqrt[3])/(3 + 3*x + x^2)^(1/3)]/3^(5/6)) + Log
[3*3^(1/3) + 3^(1/3)*x - 3*(3 + 3*x + x^2)^(1/3)]/(3*3^(1/3)) - Log[9*3^(2/3) + 6*3^(2/3)*x + 3^(2/3)*x^2 + (9
*3^(1/3) + 3*3^(1/3)*x)*(3 + 3*x + x^2)^(1/3) + 9*(3 + 3*x + x^2)^(2/3)]/(6*3^(1/3))
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fricas [A] time = 1.77, size = 156, normalized size = 0.89 \begin {gather*} \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x + 3\right )} - 3 \, {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {1}{3}} {\left (x^{2} + 6 \, x + 9\right )} + 3 \cdot 3^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} + 3 \, {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} {\left (x + 3\right )}}{x^{2}}\right ) - \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} x^{3} + 6 \cdot 3^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} {\left (x + 3\right )} - 6 \, {\left (x^{2} + 6 \, x + 9\right )} {\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (x^{3} + 18 \, x^{2} + 54 \, x + 54\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="fricas")
[Out]
1/9*3^(2/3)*log((3^(1/3)*(x + 3) - 3*(x^2 + 3*x + 3)^(1/3))/x) - 1/18*3^(2/3)*log((3^(1/3)*(x^2 + 6*x + 9) + 3
*3^(2/3)*(x^2 + 3*x + 3)^(2/3) + 3*(x^2 + 3*x + 3)^(1/3)*(x + 3))/x^2) - 1/3*3^(1/6)*arctan(1/3*3^(1/6)*(3^(1/
3)*x^3 + 6*3^(2/3)*(x^2 + 3*x + 3)^(2/3)*(x + 3) - 6*(x^2 + 6*x + 9)*(x^2 + 3*x + 3)^(1/3))/(x^3 + 18*x^2 + 54
*x + 54))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((x^2 + 3*x + 3)^(1/3)*x), x)
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maple [C] time = 14.18, size = 2320, normalized size = 13.26 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(x^2+3*x+3)^(1/3),x)
[Out]
1/9*RootOf(_Z^3-9)*ln((18684300290294655*(x^2+3*x+3)^(2/3)+19475041052229*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(
_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-1594171217561031*RootOf(_Z^3-9)*x^2-4782513652683093*RootOf(_Z^3-9)*x+518
508936864960*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3+13926825976107285*RootOf(RootOf(_Z^3-9)^2
+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2+41780477928321855*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x+41780
477928321855*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-59352506063936*RootOf(_Z^3-9)*x^3-47825136526
83093*RootOf(_Z^3-9)-170135744908815*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^
3+72915319246635*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-8346446165241*RootOf(R
ootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3+6228100096764885*(x^2+3*x+3)^(2/3)*x-460962377278
8309*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2+24305106415545*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*R
ootOf(_Z^3-9)^2*x^2-2782148721747*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2+729
15319246635*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-8346446165241*RootOf(Root
Of(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x+539492107992192*(x^2+3*x+3)^(2/3)*RootOf(_Z^3-9)^2
*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x-2076033365588295*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)*RootO
f(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-12456200193529770*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2
+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x+1618476323976576*(x^2+3*x+3)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootO
f(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-512180419198701*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x^2-307308251519220
6*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x-18684300290294655*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf
(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9))/x^3)-1/9*ln(-(-13828871318364927*(x^2+3*x+3)^(2/3)-19475041052229*RootOf(Root
Of(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-1585824771395790*RootOf(_Z^3-9)*x^2-475747431418
7370*RootOf(_Z^3-9)*x-1382249059253274*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3-186113849710480
20*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-55834154913144060*RootOf(RootOf(_Z^3-9)^2+3*_Z*Root
Of(_Z^3-9)+9*_Z^2)*x-55834154913144060*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-117777629220623*Roo
tOf(_Z^3-9)*x^3-4757474314187370*RootOf(_Z^3-9)-228560868065502*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*
_Z^2)^2*RootOf(_Z^3-9)^2*x^3+97954657742358*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-
9)^2+8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3-4609623772788309*(x^2+
3*x+3)^(2/3)*x+6228100096764885*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2+32651552580786*RootOf(RootOf(_Z^3-9)^2+3*_Z
*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+2782148721747*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^
2)*RootOf(_Z^3-9)^3*x^2+97954657742358*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*
x+8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x+539492107992192*(x^2+3*
x+3)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x+1536541257596103*(x^2+3*x+3)
^(1/3)*RootOf(_Z^3-9)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2+9219247545576618*(x^2+3*x+3)^(1/
3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x+1618476323976576*(x^2+3*x+3)^(2/3)*Roo
tOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+692011121862765*(x^2+3*x+3)^(1/3)*RootOf(_Z^
3-9)^2*x^2+4152066731176590*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x+13828871318364927*(x^2+3*x+3)^(1/3)*RootOf(Ro
otOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9))/x^3)*RootOf(_Z^3-9)-1/3*ln(-(-13828871318364927*(x^
2+3*x+3)^(2/3)-19475041052229*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-1585824
771395790*RootOf(_Z^3-9)*x^2-4757474314187370*RootOf(_Z^3-9)*x-1382249059253274*RootOf(RootOf(_Z^3-9)^2+3*_Z*R
ootOf(_Z^3-9)+9*_Z^2)*x^3-18611384971048020*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-5583415491
3144060*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x-55834154913144060*RootOf(RootOf(_Z^3-9)^2+3*_Z*R
ootOf(_Z^3-9)+9*_Z^2)-117777629220623*RootOf(_Z^3-9)*x^3-4757474314187370*RootOf(_Z^3-9)-228560868065502*RootO
f(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3+97954657742358*RootOf(RootOf(_Z^3-9)^2+3
*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2+8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2
)*RootOf(_Z^3-9)^3-4609623772788309*(x^2+3*x+3)^(2/3)*x+6228100096764885*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2+32
651552580786*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+2782148721747*RootOf(R
ootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2+97954657742358*RootOf(RootOf(_Z^3-9)^2+3*_Z*R
ootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x+8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*R
ootOf(_Z^3-9)^3*x+539492107992192*(x^2+3*x+3)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-
9)+9*_Z^2)*x+1536541257596103*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_
Z^2)*x^2+9219247545576618*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)
*x+1618476323976576*(x^2+3*x+3)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+692
011121862765*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x^2+4152066731176590*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x+1382
8871318364927*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9))/x^3)*RootO
f(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(x^2+3*x+3)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((x^2 + 3*x + 3)^(1/3)*x), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (x^2+3\,x+3\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x*(3*x + x^2 + 3)^(1/3)),x)
[Out]
int(1/(x*(3*x + x^2 + 3)^(1/3)), 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} + 3 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(x**2+3*x+3)**(1/3),x)
[Out]
Integral(1/(x*(x**2 + 3*x + 3)**(1/3)), x)
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