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

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

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Rubi [A]  time = 0.0910752, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right ) \]

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

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Rubi in Sympy [A]  time = 41.593, size = 371, normalized size = 16.13 \[ - \frac{3 \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\frac{\sqrt{3}}{3} + 1\right ) \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{- \sqrt{3} + 2} \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} + \frac{2 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} + \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{- x - 1 + \sqrt{3}}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(-2+x)/(x**3+1)**(1/2),x)

[Out]

-3*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*(sqrt(3)/3 + 1)*(x + 1)*atanh(3**(3
/4)*sqrt(-sqrt(3) + 2)*sqrt(-(-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 + 1)/(3*
sqrt((-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 - 4*sqrt(3) + 7)))/(sqrt((x + 1)
/(x + 1 + sqrt(3))**2)*(sqrt(3) + 3)*sqrt(x**3 + 1)) + 2*3**(1/4)*sqrt((x**2 - x
+ 1)/(x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(x + 1)*elliptic_f(asin((x - sqrt(
3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((x + 1)/(x + 1 + sqrt(3))**2)*
(sqrt(3) + 3)*sqrt(x**3 + 1)) + 12*3**(1/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3)
)**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_pi(4*sqrt(3) + 7, asin((-x - 1 + sqrt(
3))/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((x + 1)/(x + 1 + sqrt(3))**2)*sqrt
(-4*sqrt(3) + 7)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.282679, size = 262, normalized size = 11.39 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (\sqrt{2 i x+\sqrt{3}-i} \left (-i \sqrt{3} x+x+i \sqrt{3}+1\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}-3 i\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*(1 +
I*Sqrt[3] + x - I*Sqrt[3]*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[
2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 2*Sqrt[3]*Sqrt[I + Sqrt[3] - (2*I)*
x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(-3*I + Sqrt[3]), ArcSin[Sqrt[I + Sq
rt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((-3*I + Sqr
t[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])

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Maple [C]  time = 0.031, size = 240, normalized size = 10.4 \[ 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }-2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},-i/6\sqrt{3}+1/2,\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^3 + 1)*(x - 2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.257884, size = 59, normalized size = 2.57 \[ \frac{1}{3} \, \log \left (\frac{x^{3} + 12 \, x^{2} - 6 \, \sqrt{x^{3} + 1}{\left (x + 1\right )} - 6 \, x + 10}{x^{3} - 6 \, x^{2} + 12 \, x - 8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^3 + 1)*(x - 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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 1}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - 2\right )}\, dx \]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^3 + 1)*(x - 2)),x, algorithm="giac")

[Out]

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