3.23.91 \(\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx\)
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 = 0.21, size = 175, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {2}{\sqrt [6]{3}}+\frac {2 x}{3 \sqrt [6]{3}}+\frac {\sqrt [3]{3+3 x+x^2}}{\sqrt {3}}}{\sqrt [3]{3+3 x+x^2}}\right )}{3^{5/6}}+\frac {\log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+\left (9 \sqrt [3]{3}+3 \sqrt [3]{3} x\right ) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )}{6 \sqrt [3]{3}} \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 = 1.83, 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 = 14.56, size = 2339, normalized size = 13.37
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(2339\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(x^2+3*x+3)^(1/3),x,method=_RETURNVERBOSE)
[Out]
-1/9*ln(-(-232188818352256*RootOf(_Z^3-9)*x^3-6236446542930126*RootOf(_Z^3-9)*x^2-18709339628790378*RootOf(_Z^
3-9)*x-18684300290294655*(x^2+3*x+3)^(2/3)-18709339628790378*RootOf(_Z^3-9)-8346446165241*RootOf(RootOf(_Z^3-9
)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2-10883850860262*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3
-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2-25039338495723*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_
Z^3-9)^2*x-32651552580786*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x-5394921079921
92*(x^2+3*x+3)^(2/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2*x-539492107992192*Ro
otOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*(x^2+3*x+3)^(1/3)*x^2-3236952647953152*(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(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2-692011121862765*RootOf(_Z^3-9)^2*(x
^2+3*x+3)^(1/3)*x^2-4152066731176590*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2*x-4855428971929728*(x^2+3*x+3)^(1/3)*R
ootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)-178057518191808*RootOf(RootOf(_Z^3-9)^2+3*_Z
*RootOf(_Z^3-9)+9*_Z^2)*x^3-4782513652683093*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-143475409
58049279*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x-14347540958049279*RootOf(RootOf(_Z^3-9)^2+3*_Z*
RootOf(_Z^3-9)+9*_Z^2)-25039338495723*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-6
228100096764885*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2-6228100096764885*(x^2+3*x+3)^(2/3)*x-32651552580786*RootOf(
_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+76186956021834*RootOf(RootOf(_Z^3-9)^2+3*_Z*Root
Of(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3+58425123156687*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*R
ootOf(_Z^3-9)^2*x^3)/x^3)*RootOf(_Z^3-9)-1/3*ln(-(-232188818352256*RootOf(_Z^3-9)*x^3-6236446542930126*RootOf(
_Z^3-9)*x^2-18709339628790378*RootOf(_Z^3-9)*x-18684300290294655*(x^2+3*x+3)^(2/3)-18709339628790378*RootOf(_Z
^3-9)-8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2-10883850860262*
RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2-25039338495723*RootOf(RootOf(_Z^3-9)^
2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x-32651552580786*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(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^
2)*RootOf(_Z^3-9)^2*x-539492107992192*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*(x^2+
3*x+3)^(1/3)*x^2-3236952647953152*(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(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-
9)^2-692011121862765*RootOf(_Z^3-9)^2*(x^2+3*x+3)^(1/3)*x^2-4152066731176590*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^
2*x-4855428971929728*(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)-1780
57518191808*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3-4782513652683093*RootOf(RootOf(_Z^3-9)^2+3
*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-14347540958049279*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x-1434754
0958049279*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-25039338495723*RootOf(RootOf(_Z^3-9)^2+3*_Z*Roo
tOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-6228100096764885*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2-6228100096764885*(x
^2+3*x+3)^(2/3)*x-32651552580786*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+76186956
021834*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3+58425123156687*RootOf(RootOf(_
Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3)/x^3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+
9*_Z^2)+1/3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*ln(-(-342972057197135*RootOf(_Z^3-9)*x^3-46179
70218953550*RootOf(_Z^3-9)*x^2-13853910656860650*RootOf(_Z^3-9)*x-13828871318364927*(x^2+3*x+3)^(2/3)-13853910
656860650*RootOf(_Z^3-9)-8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*
x^2+8101702138515*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2-25039338495723*Root
Of(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x+24305106415545*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(RootOf(_Z^3-9)^2+3*_Z*Ro
otOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^2*x+539492107992192*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*Ro
otOf(_Z^3-9)*(x^2+3*x+3)^(1/3)*x^2+3236952647953152*(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(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*
_Z^2)*RootOf(_Z^3-9)^2-512180419198701*RootOf(_Z^3-9)^2*(x^2+3*x+3)^(1/3)*x^2-3073082515192206*(x^2+3*x+3)^(1/
3)*RootOf(_Z^3-9)^2*x+4855428971929728*(x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*R
ootOf(_Z^3-9)+353332887661869*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3+4757474314187370*RootOf(
RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^2+14272422942562110*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)
+9*_Z^2)*x+14272422942562110*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-25039338495723*RootOf(RootOf(
_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-4609623772788309*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2-4
609623772788309*(x^2+3*x+3)^(2/3)*x+24305106415545*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9
)+9*_Z^2)-56711914969605*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3+584251231566
87*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3)/x^3)
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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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