3.23.91 \(\int \frac {1}{x \sqrt [3]{3+3 x+x^2}} \, dx\) [2291]
Optimal. Leaf size=175 \[ -\frac {\text {ArcTan}\left (\frac {\frac {2}{\sqrt [6]{3}}+\frac {2 x}{3 \sqrt [6]{3}}+\frac {\sqrt [3]{3+3 x+x^2}}{\sqrt {3}}}{\sqrt [3]{3+3 x+x^2}}\right )}{3^{5/6}}+\frac {\log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+\left (9 \sqrt [3]{3}+3 \sqrt [3]{3} x\right ) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )}{6 \sqrt [3]{3}} \]
[Out]
-1/3*arctan((2/3*3^(5/6)+2/9*x*3^(5/6)+1/3*(x^2+3*x+3)^(1/3)*3^(1/2))/(x^2+3*x+3)^(1/3))*3^(1/6)+1/9*ln(3*3^(1
/3)+3^(1/3)*x-3*(x^2+3*x+3)^(1/3))*3^(2/3)-1/18*ln(9*3^(2/3)+6*3^(2/3)*x+3^(2/3)*x^2+(9*3^(1/3)+3*3^(1/3)*x)*(
x^2+3*x+3)^(1/3)+9*(x^2+3*x+3)^(2/3))*3^(2/3)
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Rubi [A]
time = 0.01, 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 = {766}
\begin {gather*} -\frac {\text {ArcTan}\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 \left (3^{2/3} \sqrt [3]{x^2+3 x+3}-x-3\right )}{2 \sqrt [3]{3}}-\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 766
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 [A]
time = 0.19, size = 163, normalized size = 0.93 \begin {gather*} \frac {-6 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2}{\sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}+\frac {2 x}{3 \sqrt [6]{3} \sqrt [3]{3+3 x+x^2}}\right )+\sqrt {3} \left (2 \log \left (3 \sqrt [3]{3}+\sqrt [3]{3} x-3 \sqrt [3]{3+3 x+x^2}\right )-\log \left (9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+3 \sqrt [3]{3} (3+x) \sqrt [3]{3+3 x+x^2}+9 \left (3+3 x+x^2\right )^{2/3}\right )\right )}{6\ 3^{5/6}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x*(3 + 3*x + x^2)^(1/3)),x]
[Out]
(-6*ArcTan[1/Sqrt[3] + 2/(3^(1/6)*(3 + 3*x + x^2)^(1/3)) + (2*x)/(3*3^(1/6)*(3 + 3*x + x^2)^(1/3))] + Sqrt[3]*
(2*Log[3*3^(1/3) + 3^(1/3)*x - 3*(3 + 3*x + x^2)^(1/3)] - Log[9*3^(2/3) + 6*3^(2/3)*x + 3^(2/3)*x^2 + 3*3^(1/3
)*(3 + x)*(3 + 3*x + x^2)^(1/3) + 9*(3 + 3*x + x^2)^(2/3)]))/(6*3^(5/6))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 8.38, size = 2338, normalized size = 13.36
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result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(2338\) |
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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(-(-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(_Z^3-9)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*(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+76186
956021834*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-178057518191808*RootOf(Root
Of(_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-25039338495723*RootOf(RootOf(_
Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-32651552580786*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^
3-9)+9*_Z^2)*RootOf(_Z^3-9)^3-14347540958049279*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+5842512315
6687*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-6228100096764885*(x^2+3*x+3)^(
1/3)*RootOf(_Z^3-9)^2-6228100096764885*(x^2+3*x+3)^(2/3)*x-18709339628790378*RootOf(_Z^3-9)-18684300290294655*
(x^2+3*x+3)^(2/3)-8346446165241*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^2-108
83850860262*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^2-25039338495723*RootOf(Roo
tOf(_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*Roo
tOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x-232188818352256*RootOf(_Z^3-9)*x^3-6236446542930126*RootOf(_Z^3-9)*x^2-
18709339628790378*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))/x^3)*RootOf(_Z^3-9)-1/3*ln(-(-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(_Z^3-9)*RootOf(RootOf(_Z^3-9)^2+3*_Z*Root
Of(_Z^3-9)+9*_Z^2)*(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*RootO
f(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x+76186956021834*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_
Z^3-9)^3*x^3-178057518191808*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3-4782513652683093*RootOf(R
ootOf(_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-25039338495723*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2-32651552580786
*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3-14347540958049279*RootOf(RootOf(_Z^3-9)^
2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)+58425123156687*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3
-9)^2*x^3-6228100096764885*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2-6228100096764885*(x^2+3*x+3)^(2/3)*x-18709339628
790378*RootOf(_Z^3-9)-18684300290294655*(x^2+3*x+3)^(2/3)-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-3265155
2580786*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x-232188818352256*RootOf(_Z^3-9)*
x^3-6236446542930126*RootOf(_Z^3-9)*x^2-18709339628790378*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))/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((-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(_Z^3-9)*RootOf(RootOf(_
Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*(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+56711914969605*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9
)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-353332887661869*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3-4757474
314187370*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+25039338495723*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9
)^2-24305106415545*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3-14272422942562110*Root
Of(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)-58425123156687*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_
Z^2)^2*RootOf(_Z^3-9)^2*x^3+4609623772788309*(x^2+3*x+3)^(1/3)*RootOf(_Z^3-9)^2+4609623772788309*(x^2+3*x+3)^(
2/3)*x+13853910656860650*RootOf(_Z^3-9)+1382887...
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [A]
time = 1.36, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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