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 = 0.177987, 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 \[ -\frac{\log \left (-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}}{\sqrt{3}}\right )}{\sqrt{3} (a+1)^{2/3} \sqrt [3]{b}} \]

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))

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Rubi in Sympy [A]  time = 28.7938, size = 117, normalized size = 0.94 \[ \frac{\log{\left (\sqrt [3]{b} x + \sqrt [3]{a + 1} \right )}}{3 \sqrt [3]{b} \left (a + 1\right )^{\frac{2}{3}}} - \frac{\log{\left (b^{\frac{2}{3}} x^{2} - \sqrt [3]{b} x \sqrt [3]{a + 1} + \left (a + 1\right )^{\frac{2}{3}} \right )}}{6 \sqrt [3]{b} \left (a + 1\right )^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + 1}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{b} \left (a + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**3+a+1),x)

[Out]

log(b**(1/3)*x + (a + 1)**(1/3))/(3*b**(1/3)*(a + 1)**(2/3)) - log(b**(2/3)*x**2
- b**(1/3)*x*(a + 1)**(1/3) + (a + 1)**(2/3))/(6*b**(1/3)*(a + 1)**(2/3)) - sqr
t(3)*atan(sqrt(3)*(-2*b**(1/3)*x/(3*(a + 1)**(1/3)) + 1/3))/(3*b**(1/3)*(a + 1)*
*(2/3))

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Mathematica [A]  time = 0.0604058, 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 = 0.009, size = 105, normalized size = 0.8 \[{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{1+a}{b}}} \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{1+a}{b}}}+ \left ({\frac{1+a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{1+a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{1+a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

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-x*((1+a
)/b)^(1/3)+((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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 [A]  time = 0.216612, size = 169, normalized size = 1.35 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{2}{3}} x^{2} - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}}{\left (a + 1\right )} x + a^{2} + 2 \, a + 1\right ) - 2 \, \sqrt{3} \log \left (\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}} x + a + 1\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}} x - \sqrt{3}{\left (a + 1\right )}}{3 \,{\left (a + 1\right )}}\right )\right )}}{18 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*log(((a^2 + 2*a + 1)*b)^(2/3)*x^2 - ((a^2 + 2*a + 1)*b)^(
1/3)*(a + 1)*x + a^2 + 2*a + 1) - 2*sqrt(3)*log(((a^2 + 2*a + 1)*b)^(1/3)*x + a
+ 1) - 6*arctan(1/3*(2*sqrt(3)*((a^2 + 2*a + 1)*b)^(1/3)*x - sqrt(3)*(a + 1))/(a
+ 1)))/((a^2 + 2*a + 1)*b)^(1/3)

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Sympy [A]  time = 0.620472, size = 32, normalized size = 0.26 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log{\left (3 t a + 3 t + x \right )} \right )\right )} \]

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/XCAS [A]  time = 0.248967, size = 193, normalized size = 1.54 \[ \frac{{\left (-a b^{2} - b^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b + \sqrt{3} b} + \frac{{\left (-a b^{2} - b^{2}\right )}^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}} + \left (-\frac{a + 1}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b + b\right )}} - \frac{\left (-\frac{a + 1}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a + 1}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a + 1\right )}} \]

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