3.103 \(\int \frac{1+x}{\left (1-\sqrt{3}+x\right ) \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=145 \[ \frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

[Out]

-(ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]/Sqrt[-3 + 2*Sqrt[3]]) +
(Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[Arc
Sin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)
/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

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Rubi [A]  time = 0.378841, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]/Sqrt[-3 + 2*Sqrt[3]]) +
(Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[Arc
Sin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)
/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

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Rubi in Sympy [A]  time = 41.3084, size = 226, normalized size = 1.56 \[ - \frac{3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{\left (- \sqrt{3} + 2\right ) \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{\sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt{x^{3} + 1}} + \frac{3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*(x + 1)*atanh((-sqrt(3) + 2)
*sqrt(-(-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 + 1)/sqrt((-x - 1 + sqrt(3))**
2/(x + 1 + sqrt(3))**2 - 4*sqrt(3) + 7))/(3*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*s
qrt(-sqrt(3) + 2)*sqrt(x**3 + 1)) + 3**(3/4)*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))/(3*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.454259, size = 267, normalized size = 1.84 \[ -\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (2 i \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 i \sqrt{3}}{-3+(2+i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}-i} \left (\left (\sqrt{3}+(-2-i)\right ) x-i \sqrt{3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (-3+(2+i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + x)/((1 - Sqrt[3] + 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
+ 2*I) - I*Sqrt[3] + ((-2 - 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*I)*Sqrt[I + Sqrt
[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(-3 + (2 + I)*Sqrt[3
]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sq
rt[3])]))/((-3 + (2 + I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])

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Maple [B]  time = 0.033, size = 245, normalized size = 1.7 \[ 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }-2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},-1/3\, \left ( -3/2+i/2\sqrt{3} \right ) \sqrt{3},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(1+x-3^(1/2))/(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),-1/3*(-3/2+1/2*I*3^(1/2))*3^(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. \[ \int \frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x + 1}{\sqrt{x^{3} + 1}{\left (x - \sqrt{3} + 1\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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