3.27 \(\int \frac{1}{a+d x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0630292, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

Int[(a + d*x^3)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0266957, size = 89, normalized size = 0.77 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + d*x^3)^(-1),x]

[Out]

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

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Maple [A]  time = 0.004, size = 91, normalized size = 0.8 \begin{align*}{\frac{1}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{a}{d}}} \right ) \left ({\frac{a}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{d}}}x+ \left ({\frac{a}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{d}}}}}}-1 \right ) } \right ) \left ({\frac{a}{d}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x^3+a),x)

[Out]

1/3/d/(a/d)^(2/3)*ln(x+(a/d)^(1/3))-1/6/d/(a/d)^(2/3)*ln(x^2-(a/d)^(1/3)*x+(a/d)^(2/3))+1/3/d/(a/d)^(2/3)*3^(1
/2)*arctan(1/3*3^(1/2)*(2/(a/d)^(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/(d*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.22155, size = 749, normalized size = 6.51 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a d \sqrt{-\frac{\left (a^{2} d\right )^{\frac{1}{3}}}{d}} \log \left (\frac{2 \, a d x^{3} - 3 \, \left (a^{2} d\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a d x^{2} + \left (a^{2} d\right )^{\frac{2}{3}} x - \left (a^{2} d\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} d\right )^{\frac{1}{3}}}{d}}}{d x^{3} + a}\right ) - \left (a^{2} d\right )^{\frac{2}{3}} \log \left (a d x^{2} - \left (a^{2} d\right )^{\frac{2}{3}} x + \left (a^{2} d\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} d\right )^{\frac{2}{3}} \log \left (a d x + \left (a^{2} d\right )^{\frac{2}{3}}\right )}{6 \, a^{2} d}, \frac{6 \, \sqrt{\frac{1}{3}} a d \sqrt{\frac{\left (a^{2} d\right )^{\frac{1}{3}}}{d}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} d\right )^{\frac{2}{3}} x - \left (a^{2} d\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} d\right )^{\frac{1}{3}}}{d}}}{a^{2}}\right ) - \left (a^{2} d\right )^{\frac{2}{3}} \log \left (a d x^{2} - \left (a^{2} d\right )^{\frac{2}{3}} x + \left (a^{2} d\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} d\right )^{\frac{2}{3}} \log \left (a d x + \left (a^{2} d\right )^{\frac{2}{3}}\right )}{6 \, a^{2} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.144555, size = 20, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} d - 1, \left ( t \mapsto t \log{\left (3 t a + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15588, size = 151, normalized size = 1.31 \begin{align*} -\frac{\left (-\frac{a}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} + \frac{\sqrt{3} \left (-a d^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{d}\right )^{\frac{1}{3}}}\right )}{3 \, a d} + \frac{\left (-a d^{2}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{d}\right )^{\frac{1}{3}} + \left (-\frac{a}{d}\right )^{\frac{2}{3}}\right )}{6 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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