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

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

[Out]

-(ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]/Sqrt[3 + 2*Sqrt[3]]) + (Sq
rt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin
[(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 = 0.356485, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right )}{\sqrt{3+2 \sqrt{3}}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]/Sqrt[3 + 2*Sqrt[3]]) + (Sq
rt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin
[(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 = 13.2164, size = 78, normalized size = 0.54 \[ \frac{2 \tilde{\infty } \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

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]

2*zoo*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*(x + 1)*elliptic_f(asin((x - sqr
t(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((x + 1)/(x + 1 + sqrt(3))**2
)*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.532937, size = 269, normalized size = 1.86 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}-i} \left (\left ((1+2 i)+i \sqrt{3}\right ) x-\sqrt{3}-(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]

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]*((-2
- I) - Sqrt[3] + ((1 + 2*I) + 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[I + Sqrt[3] -
(2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), Ar
cSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])
]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])

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Maple [B]  time = 0.034, 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. \[ \int \frac{x + 1}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt{3}\right )}\, dx \]

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 + 1 + sqrt(3))), 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)