3.74 \(\int \frac{1+x}{(2-x) \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right ) \]

[Out]

(2*ArcTanh[(1 + x)^2/(3*Sqrt[1 + x^3])])/3

________________________________________________________________________________________

Rubi [A]  time = 0.0592311, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2138, 206} \[ \frac{2}{3} \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)/((2 - x)*Sqrt[1 + x^3]),x]

[Out]

(2*ArcTanh[(1 + x)^2/(3*Sqrt[1 + x^3])])/3

Rule 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-2*e)/d, Subst[Int
[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1+x}{(2-x) \sqrt{1+x^3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{9-x^2} \, dx,x,\frac{(1+x)^2}{\sqrt{1+x^3}}\right )\\ &=\frac{2}{3} \tanh ^{-1}\left (\frac{(1+x)^2}{3 \sqrt{1+x^3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0093062, size = 46, normalized size = 2. \[ \frac{1}{3} \log \left (\frac{(x+1)^2}{\sqrt{x^3+1}}+3\right )-\frac{1}{3} \log \left (3-\frac{(x+1)^2}{\sqrt{x^3+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/((2 - x)*Sqrt[1 + x^3]),x]

[Out]

-Log[3 - (1 + x)^2/Sqrt[1 + x^3]]/3 + Log[3 + (1 + x)^2/Sqrt[1 + x^3]]/3

________________________________________________________________________________________

Maple [C]  time = 0.02, size = 240, normalized size = 10.4 \begin{align*} -2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},1/2-i/6\sqrt{3},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(2-x)/(x^3+1)^(1/2),x)

[Out]

-2*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((
x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-
3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+2*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/
2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*
EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),1/2-1/6*I*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/
2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(2-x)/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

-integrate((x + 1)/(sqrt(x^3 + 1)*(x - 2)), x)

________________________________________________________________________________________

Fricas [B]  time = 1.99485, size = 117, normalized size = 5.09 \begin{align*} \frac{1}{3} \, \log \left (\frac{x^{3} + 12 \, x^{2} + 6 \, \sqrt{x^{3} + 1}{\left (x + 1\right )} - 6 \, x + 10}{x^{3} - 6 \, x^{2} + 12 \, x - 8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(2-x)/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

1/3*log((x^3 + 12*x^2 + 6*sqrt(x^3 + 1)*(x + 1) - 6*x + 10)/(x^3 - 6*x^2 + 12*x - 8))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{x \sqrt{x^{3} + 1} - 2 \sqrt{x^{3} + 1}}\, dx - \int \frac{1}{x \sqrt{x^{3} + 1} - 2 \sqrt{x^{3} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(2-x)/(x**3+1)**(1/2),x)

[Out]

-Integral(x/(x*sqrt(x**3 + 1) - 2*sqrt(x**3 + 1)), x) - Integral(1/(x*sqrt(x**3 + 1) - 2*sqrt(x**3 + 1)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(2-x)/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

integrate(-(x + 1)/(sqrt(x^3 + 1)*(x - 2)), x)