3.5.10 \(\int \frac {1}{x^6 (1+x^5+x^{10})} \, dx\) [410]

Optimal. Leaf size=48 \[ -\frac {1}{5 x^5}-\frac {\tan ^{-1}\left (\frac {1+2 x^5}{\sqrt {3}}\right )}{5 \sqrt {3}}-\log (x)+\frac {1}{10} \log \left (1+x^5+x^{10}\right ) \]

[Out]

-1/5/x^5-ln(x)+1/10*ln(x^10+x^5+1)-1/15*arctan(1/3*(2*x^5+1)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1371, 723, 814, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 x^5+1}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {1}{5 x^5}+\frac {1}{10} \log \left (x^{10}+x^5+1\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(1 + x^5 + x^10)),x]

[Out]

-1/5*1/x^5 - ArcTan[(1 + 2*x^5)/Sqrt[3]]/(5*Sqrt[3]) - Log[x] + Log[1 + x^5 + x^10]/10

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 723

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m
+ 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c
*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (1+x^5+x^{10}\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x+x^2\right )} \, dx,x,x^5\right )\\ &=-\frac {1}{5 x^5}+\frac {1}{5} \text {Subst}\left (\int \frac {-1-x}{x \left (1+x+x^2\right )} \, dx,x,x^5\right )\\ &=-\frac {1}{5 x^5}+\frac {1}{5} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {x}{1+x+x^2}\right ) \, dx,x,x^5\right )\\ &=-\frac {1}{5 x^5}-\log (x)+\frac {1}{5} \text {Subst}\left (\int \frac {x}{1+x+x^2} \, dx,x,x^5\right )\\ &=-\frac {1}{5 x^5}-\log (x)-\frac {1}{10} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^5\right )+\frac {1}{10} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^5\right )\\ &=-\frac {1}{5 x^5}-\log (x)+\frac {1}{10} \log \left (1+x^5+x^{10}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^5\right )\\ &=-\frac {1}{5 x^5}-\frac {\tan ^{-1}\left (\frac {1+2 x^5}{\sqrt {3}}\right )}{5 \sqrt {3}}-\log (x)+\frac {1}{10} \log \left (1+x^5+x^{10}\right )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.03, size = 208, normalized size = 4.33 \begin {gather*} \frac {1}{30} \left (-\frac {6}{x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-30 \log (x)+3 \log \left (1+x+x^2\right )+6 \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^3-\text {$\#$1}^4+\text {$\#$1}^5-\text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2-3 \log (x-\text {$\#$1}) \text {$\#$1}^3+2 \log (x-\text {$\#$1}) \text {$\#$1}^4+\log (x-\text {$\#$1}) \text {$\#$1}^5-4 \log (x-\text {$\#$1}) \text {$\#$1}^6+4 \log (x-\text {$\#$1}) \text {$\#$1}^7}{-1+3 \text {$\#$1}^2-4 \text {$\#$1}^3+5 \text {$\#$1}^4-7 \text {$\#$1}^6+8 \text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(1 + x^5 + x^10)),x]

[Out]

(-6/x^5 + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 30*Log[x] + 3*Log[1 + x + x^2] + 6*RootSum[1 - #1 + #1^3 - #1^
4 + #1^5 - #1^7 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1 + Log[x - #1]*#1^2 - 3*Log[x - #1]*#1^3 + 2*Log[x -
#1]*#1^4 + Log[x - #1]*#1^5 - 4*Log[x - #1]*#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^4 - 7*#1^6
+ 8*#1^7) & ])/30

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Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 138, normalized size = 2.88

method	result	size
risch	\(-\frac {1}{5 x^{5}}-\ln \left (x \right )+\frac {\ln \left (x^{10}+x^{5}+1\right )}{10}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{5}+\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{15}\)	\(38\)
default	\(\frac {\ln \left (x^{2}+x +1\right )}{10}+\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{15}+\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{6}\right ) \ln \left (2 x^{4}+\left (-1-i \sqrt {3}\right ) x^{3}+\left (-1+i \sqrt {3}\right ) x^{2}+2 x -1-i \sqrt {3}\right )}{5}+\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) \ln \left (2 x^{4}+\left (-1+i \sqrt {3}\right ) x^{3}+\left (-1-i \sqrt {3}\right ) x^{2}+2 x -1+i \sqrt {3}\right )}{5}-\frac {1}{5 x^{5}}-\ln \left (x \right )\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^10+x^5+1),x,method=_RETURNVERBOSE)

[Out]

1/10*ln(x^2+x+1)+1/15*arctan(1/3*(2*x+1)*3^(1/2))*3^(1/2)+1/5*(1/2+1/6*I*3^(1/2))*ln(2*x^4+(-1-I*3^(1/2))*x^3+
(-1+I*3^(1/2))*x^2+2*x-1-I*3^(1/2))+1/5*(1/2-1/6*I*3^(1/2))*ln(2*x^4+(-1+I*3^(1/2))*x^3+(-1-I*3^(1/2))*x^2+2*x
-1+I*3^(1/2))-1/5/x^5-ln(x)

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Maxima [A]
time = 0.51, size = 41, normalized size = 0.85 \begin {gather*} -\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - \frac {1}{5 \, x^{5}} + \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) - \frac {1}{5} \, \log \left (x^{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^10+x^5+1),x, algorithm="maxima")

[Out]

-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/5/x^5 + 1/10*log(x^10 + x^5 + 1) - 1/5*log(x^5)

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Fricas [A]
time = 0.36, size = 49, normalized size = 1.02 \begin {gather*} -\frac {2 \, \sqrt {3} x^{5} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - 3 \, x^{5} \log \left (x^{10} + x^{5} + 1\right ) + 30 \, x^{5} \log \left (x\right ) + 6}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^10+x^5+1),x, algorithm="fricas")

[Out]

-1/30*(2*sqrt(3)*x^5*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 3*x^5*log(x^10 + x^5 + 1) + 30*x^5*log(x) + 6)/x^5

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Sympy [A]
time = 0.08, size = 48, normalized size = 1.00 \begin {gather*} - \log {\left (x \right )} + \frac {\log {\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{5}}{3} + \frac {\sqrt {3}}{3} \right )}}{15} - \frac {1}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(x**10+x**5+1),x)

[Out]

-log(x) + log(x**10 + x**5 + 1)/10 - sqrt(3)*atan(2*sqrt(3)*x**5/3 + sqrt(3)/3)/15 - 1/(5*x**5)

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Giac [A]
time = 3.32, size = 45, normalized size = 0.94 \begin {gather*} -\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) + \frac {x^{5} - 1}{5 \, x^{5}} + \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^10+x^5+1),x, algorithm="giac")

[Out]

-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) + 1/5*(x^5 - 1)/x^5 + 1/10*log(x^10 + x^5 + 1) - log(abs(x))

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Mupad [B]
time = 1.37, size = 41, normalized size = 0.85 \begin {gather*} \frac {\ln \left (x^{10}+x^5+1\right )}{10}-\ln \left (x\right )-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^5}{3}+\frac {\sqrt {3}}{3}\right )}{15}-\frac {1}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^5 + x^10 + 1)),x)

[Out]

log(x^5 + x^10 + 1)/10 - log(x) - (3^(1/2)*atan(3^(1/2)/3 + (2*3^(1/2)*x^5)/3))/15 - 1/(5*x^5)

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