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

Optimal. Leaf size=342 \[ \frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \tan ^{-1}\left (\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}}}{\sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}}}\right )}{\sqrt{26} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{2 (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{4 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (97-56 \sqrt{3};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

((1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(Sqrt[13/2]*Sqrt[(1 + x)
/(1 + Sqrt[3] + x)^2])/Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]])/(Sqrt[26]*Sqrt[
(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3]) + (2*(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[2 - Sqrt[3]]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-
1 - x^3]) + (4*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticP
i[97 - 56*Sqrt[3], -ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]]
)/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3])

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Rubi [A]  time = 1.35035, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \tan ^{-1}\left (\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}}}{\sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}}}\right )}{\sqrt{26} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{2 (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{4 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (97-56 \sqrt{3};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

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

[Out]

((1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(Sqrt[13/2]*Sqrt[(1 + x)
/(1 + Sqrt[3] + x)^2])/Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]])/(Sqrt[26]*Sqrt[
(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3]) + (2*(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[2 - Sqrt[3]]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-
1 - x^3]) + (4*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticP
i[97 - 56*Sqrt[3], -ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]]
)/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[-1 - x^3])

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Rubi in Sympy [A]  time = 93.9631, size = 376, normalized size = 1.1 \[ \frac{\sqrt{26} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{\sqrt{26} \cdot 3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{6 \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{26 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (\sqrt{3} + 2\right ) \sqrt{- x^{3} - 1}} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \Pi \left (\frac{\left (\sqrt{3} + 2\right )^{2}}{\left (-2 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 2\right ) \sqrt{\sqrt{3} + 2} \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(26)*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*(x + 1)*atanh(sqrt(26)*3**(3/
4)*sqrt(1 - (x + 1 + sqrt(3))**2/(-x - 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)/(6*sqr
t(4*sqrt(3) + 7 + (x + 1 + sqrt(3))**2/(-x - 1 + sqrt(3))**2)))/(26*sqrt((-x - 1
)/(x - sqrt(3) + 1)**2)*sqrt(-x**3 - 1)) + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x - s
qrt(3) + 1)**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_f(asin((x + 1 + sqrt(3))/(x
- sqrt(3) + 1)), -7 + 4*sqrt(3))/(3*sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*(sqrt(3)
+ 2)*sqrt(-x**3 - 1)) - 4*3**(1/4)*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*(x
+ 1)*elliptic_pi((sqrt(3) + 2)**2/(-2 + sqrt(3))**2, asin((x + 1 + sqrt(3))/(-x
- 1 + sqrt(3))), -7 + 4*sqrt(3))/(sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*(-sqrt(3)
+ 2)*sqrt(sqrt(3) + 2)*sqrt(4*sqrt(3) + 7)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.0939093, size = 130, normalized size = 0.38 \[ -\frac{4 \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{7 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )}{\left (\sqrt{3}+7 i\right ) \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sq
rt[3])/(7*I + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (
2*Sqrt[3])/(3*I + Sqrt[3])])/((7*I + Sqrt[3])*Sqrt[-1 - x^3])

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Maple [A]  time = 0.055, size = 133, normalized size = 0.4 \[{\frac{-{\frac{2\,i}{3}}\sqrt{3}}{{\frac{7}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{7}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3+x)/(-x^3-1)^(1/2),x)

[Out]

-2/3*I*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)
))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)/(7/2+1/2*I*3^(1
/2))*EllipticPi(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(7
/2+1/2*I*3^(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. \[ \int \frac{1}{\sqrt{-x^{3} - 1}{\left (x + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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