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

Optimal. Leaf size=382 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}}}{2 \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}}}\right )}{2 \sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (4+\sqrt{3}\right ) \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{1}{169} \left (553+304 \sqrt{3}\right );-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

-((1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[7]*Sqrt[(1 - x)/
(1 + Sqrt[3] - x)^2])/(2*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2])])/(2*Sqrt[7]*S
qrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (2*Sqrt[2 + Sqrt[3]]*(1 - x)*S
qrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + S
qrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(4 + Sqrt[3])*Sqrt[(1 - x)/(1 + Sqrt[3]
- x)^2]*Sqrt[1 - x^3]) + (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)
/(1 + Sqrt[3] - x)^2]*EllipticPi[(553 + 304*Sqrt[3])/169, -ArcSin[(1 - Sqrt[3] -
x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(13*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*S
qrt[1 - x^3])

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Rubi [A]  time = 1.50548, antiderivative size = 382, 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 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}}}{2 \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}}}\right )}{2 \sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (4+\sqrt{3}\right ) \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{1}{169} \left (553+304 \sqrt{3}\right );-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

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]*ArcTanh[(Sqrt[7]*Sqrt[(1 - x)/
(1 + Sqrt[3] - x)^2])/(2*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2])])/(2*Sqrt[7]*S
qrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) - (2*Sqrt[2 + Sqrt[3]]*(1 - x)*S
qrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + S
qrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(4 + Sqrt[3])*Sqrt[(1 - x)/(1 + Sqrt[3]
- x)^2]*Sqrt[1 - x^3]) + (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)
/(1 + Sqrt[3] - x)^2]*EllipticPi[(553 + 304*Sqrt[3])/169, -ArcSin[(1 - Sqrt[3] -
x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(13*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*S
qrt[1 - x^3])

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Rubi in Sympy [A]  time = 93.387, size = 376, normalized size = 0.98 \[ - \frac{\sqrt{7} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{7} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2}}{6 \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{14 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 4\right ) \sqrt{- x^{3} + 1}} + \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (\sqrt{3} + 4\right )^{2}}{\left (-4 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 4\right ) \left (\sqrt{3} + 4\right ) \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(7)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(-x + 1)*atanh(3**(3/4)*sqrt
(7)*sqrt(1 - (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)/(6*s
qrt(-4*sqrt(3) + 7 + (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)))/(14*sqrt((-x
+ 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)) - 2*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(3) + 4)*sqrt(-x**3 + 1)) + 4*3**(1/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3
))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_pi((sqrt(3) + 4)**2/(-4 + sqrt(3))**
2, 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) + 4)*(sqrt(3) + 4)*sqrt(-x**
3 + 1))

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Mathematica [C]  time = 0.0937006, size = 128, normalized size = 0.34 \[ -\frac{4 \sqrt{2} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )}{\left (\sqrt{3}+5 i\right ) \sqrt{1-x^3}} \]

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*
Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))]
, (2*Sqrt[3])/(-3*I + Sqrt[3])])/((5*I + Sqrt[3])*Sqrt[1 - x^3])

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Maple [A]  time = 0.072, size = 133, normalized size = 0.4 \[{\frac{-{\frac{2\,i}{3}}\sqrt{3}}{{\frac{5}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{5}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

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)/(5/2+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)/
(5/2+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)