3.14.29 \(\int \frac {1}{x^3 (a+b x^6)} \, dx\) [1329]

Optimal. Leaf size=133 \[ -\frac {1}{2 a x^2}+\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {281, 331, 298, 31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}-\frac {1}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^6)),x]

[Out]

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

Rule 31

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

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 281

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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 203, normalized size = 1.53 \begin {gather*} \frac {-6 \sqrt [3]{a}+2 \sqrt {3} \sqrt [3]{b} x^2 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \sqrt [3]{b} x^2 \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 a^{4/3} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^6)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 112, normalized size = 0.84

method	result	size
risch	\(-\frac {1}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{4} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-7 a^{4} \textit {\_R}^{3}+6 b \right ) x^{2}-a^{3} \textit {\_R}^{2}\right )\right )}{6}\)	\(55\)
default	\(-\frac {1}{2 a \,x^{2}}-\frac {b \left (-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 a}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x^6+a),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 112, normalized size = 0.84 \begin {gather*} -\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 \, a \left (\frac {a}{b}\right )^{\frac {1}{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 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {1}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 115, normalized size = 0.86 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{12 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^6+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.11, size = 34, normalized size = 0.26 \begin {gather*} \operatorname {RootSum} {\left (216 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (\frac {36 t^{2} a^{3}}{b} + x^{2} \right )} \right )\right )} - \frac {1}{2 a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**6+a),x)

[Out]

RootSum(216*_t**3*a**4 - b, Lambda(_t, _t*log(36*_t**2*a**3/b + x**2))) - 1/(2*a*x**2)

________________________________________________________________________________________

Giac [A]
time = 2.72, size = 127, normalized size = 0.95 \begin {gather*} \frac {b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, a^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{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^{2} b} - \frac {\left (-a b^{2}\right )^{\frac {2}{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^{2} b} - \frac {1}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^6+a),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.17, size = 116, normalized size = 0.87 \begin {gather*} \frac {b^{1/3}\,\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{6\,a^{4/3}}-\frac {1}{2\,a\,x^2}+\frac {b^{1/3}\,\ln \left (a^4\,b^6+a^{11/3}\,b^{19/3}\,x^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,a^{4/3}}-\frac {b^{1/3}\,\ln \left (a^4\,b^6-a^{11/3}\,b^{19/3}\,x^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,a^{4/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^6)),x)

[Out]

(b^(1/3)*log(a^(1/3) + b^(1/3)*x^2))/(6*a^(4/3)) - 1/(2*a*x^2) + (b^(1/3)*log(a^4*b^6 + a^(11/3)*b^(19/3)*x^2*
((3^(1/2)*1i)/2 - 1/2))*((3^(1/2)*1i)/2 - 1/2))/(6*a^(4/3)) - (b^(1/3)*log(a^4*b^6 - a^(11/3)*b^(19/3)*x^2*((3
^(1/2)*1i)/2 + 1/2))*((3^(1/2)*1i)/2 + 1/2))/(6*a^(4/3))

________________________________________________________________________________________