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

Optimal. Leaf size=125 \[ -\frac{\log \left (-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}}{\sqrt{3}}\right )}{\sqrt{3} (a+1)^{2/3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.085638, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}}{\sqrt{3}}\right )}{\sqrt{3} (a+1)^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + a + b*x^3)^(-1),x]

[Out]

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

Mathematica [A]  time = 0.0341274, size = 101, normalized size = 0.81 \[ \frac{-\log \left (-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )+2 \log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}-1}{\sqrt{3}}\right )}{6 (a+1)^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + a + b*x^3)^(-1),x]

[Out]

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

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Maple [A]  time = 0.005, size = 105, normalized size = 0.8 \begin{align*}{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{1+a}{b}}} \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{1+a}{b}}}x+ \left ({\frac{1+a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{1+a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.55975, size = 1230, normalized size = 9.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.264982, size = 32, normalized size = 0.26 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log{\left (3 t a + 3 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**3*(27*a**2*b + 54*a*b + 27*b) - 1, Lambda(_t, _t*log(3*_t*a + 3*_t + x)))

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Giac [A]  time = 1.14399, size = 193, normalized size = 1.54 \begin{align*} \frac{{\left (-a b^{2} - b^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b + \sqrt{3} b} + \frac{{\left (-a b^{2} - b^{2}\right )}^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}} + \left (-\frac{a + 1}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b + b\right )}} - \frac{\left (-\frac{a + 1}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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