3.1973 \(\int \frac{1}{(a+\frac{b}{x^3}) x^3} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0509691, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {263, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)*x^3),x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 31

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

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]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0118458, size = 89, normalized size = 0.77 \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{6 \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)*x^3),x]

[Out]

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

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Maple [A]  time = 0.001, size = 91, normalized size = 0.8 \begin{align*}{\frac{1}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^3)/x^3,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50951, size = 749, normalized size = 6.51 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac{1}{3}} b x - b^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a b^{2}\right )^{\frac{2}{3}} x - \left (a b^{2}\right )^{\frac{1}{3}} b\right )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{a x^{3} + b}\right ) - \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac{2}{3}} x + \left (a b^{2}\right )^{\frac{1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a b^{2}\right )^{\frac{2}{3}} x - \left (a b^{2}\right )^{\frac{1}{3}} b\right )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{b^{2}}\right ) - \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac{2}{3}} x + \left (a b^{2}\right )^{\frac{1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.183262, size = 20, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a b^{2} - 1, \left ( t \mapsto t \log{\left (3 t b + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a*b**2 - 1, Lambda(_t, _t*log(3*_t*b + x)))

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Giac [A]  time = 1.13651, size = 151, normalized size = 1.31 \begin{align*} -\frac{\left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a b} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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