3.1325 \(\int \frac {x}{a+b x^6} \, dx\)

Optimal. Leaf size=123 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{b}} \]

[Out]

1/6*ln(a^(1/3)+b^(1/3)*x^2)/a^(2/3)/b^(1/3)-1/12*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/a^(2/3)/b^(1/3)-1
/6*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/a^(2/3)/b^(1/3)*3^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {275, 200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^6),x]

[Out]

-ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(2*Sqrt[3]*a^(2/3)*b^(1/3)) + Log[a^(1/3) + b^(1/3)*x^2]/
(6*a^(2/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(12*a^(2/3)*b^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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 634

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]

Rubi steps

\begin {align*} \int \frac {x}{a+b x^6} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 a^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{6 a^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{a}}-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 a^{2/3} \sqrt [3]{b}}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{2 a^{2/3} \sqrt [3]{b}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 154, normalized size = 1.25 \[ -\frac {-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )}{12 a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^6),x]

[Out]

-1/12*(2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*Sqrt[3]*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] -
2*Log[a^(1/3) + b^(1/3)*x^2] + Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + Log[a^(1/3) + Sqrt[3]
*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(a^(2/3)*b^(1/3))

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fricas [A]  time = 0.99, size = 313, normalized size = 2.54 \[ \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{6} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{2} - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{4} + \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{6} + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{12 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{12 \, a^{2} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^6+a),x, algorithm="fricas")

[Out]

[1/12*(3*sqrt(1/3)*a*b*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^6 - 3*(a^2*b)^(1/3)*a*x^2 - a^2 + 3*sqrt(1/3)*(2*a*
b*x^4 + (a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^6 + a)) - (a^2*b)^(2/3)*log(a*b*x^4
- (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*x^2 + (a^2*b)^(2/3)))/(a^2*b), 1/12*(6*sqrt(1
/3)*a*b*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a
^2) - (a^2*b)^(2/3)*log(a*b*x^4 - (a^2*b)^(2/3)*x^2 + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*x^2 + (a^2*b)
^(2/3)))/(a^2*b)]

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giac [A]  time = 0.19, size = 118, normalized size = 0.96 \[ -\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, a} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, a b} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^6+a),x, algorithm="giac")

[Out]

-1/6*(-a/b)^(1/3)*log(abs(x^2 - (-a/b)^(1/3)))/a + 1/6*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/
b)^(1/3))/(-a/b)^(1/3))/(a*b) + 1/12*(-a*b^2)^(1/3)*log(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b)

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maple [A]  time = 0.00, size = 97, normalized size = 0.79 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \left (\frac {a}{b}\right )^{\frac {2}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^6+a),x)

[Out]

1/6/b/(a/b)^(2/3)*ln(x^2+(a/b)^(1/3))-1/12/b/(a/b)^(2/3)*ln(x^4-(a/b)^(1/3)*x^2+(a/b)^(2/3))+1/6/b/(a/b)^(2/3)
*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^2-1))

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maxima [A]  time = 2.34, size = 104, normalized size = 0.85 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^6+a),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - (a/b)^(1/3))/(a/b)^(1/3))/(b*(a/b)^(2/3)) - 1/12*log(x^4 - x^2*(a/b)^(
1/3) + (a/b)^(2/3))/(b*(a/b)^(2/3)) + 1/6*log(x^2 + (a/b)^(1/3))/(b*(a/b)^(2/3))

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mupad [B]  time = 0.24, size = 105, normalized size = 0.85 \[ \frac {\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{6\,a^{2/3}\,b^{1/3}}+\frac {\ln \left (6\,b^4\,x^2+3\,a^{1/3}\,b^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12\,a^{2/3}\,b^{1/3}}-\frac {\ln \left (6\,b^4\,x^2-3\,a^{1/3}\,b^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12\,a^{2/3}\,b^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^6),x)

[Out]

log(a^(1/3) + b^(1/3)*x^2)/(6*a^(2/3)*b^(1/3)) + (log(6*b^4*x^2 + 3*a^(1/3)*b^(11/3)*(3^(1/2)*1i - 1))*(3^(1/2
)*1i - 1))/(12*a^(2/3)*b^(1/3)) - (log(6*b^4*x^2 - 3*a^(1/3)*b^(11/3)*(3^(1/2)*1i + 1))*(3^(1/2)*1i + 1))/(12*
a^(2/3)*b^(1/3))

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sympy [A]  time = 0.24, size = 22, normalized size = 0.18 \[ \operatorname {RootSum} {\left (216 t^{3} a^{2} b - 1, \left (t \mapsto t \log {\left (6 t a + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**6+a),x)

[Out]

RootSum(216*_t**3*a**2*b - 1, Lambda(_t, _t*log(6*_t*a + x**2)))

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