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

Optimal. Leaf size=376 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}}}{2 \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}}}\right )}{2 \sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2 \sqrt{62-35 \sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{13 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{1}{169} \left (553+304 \sqrt{3}\right );-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[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[62 - 35*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]])/(13*3^(1/4)*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]*Sqr
t[-1 + x^3])

_______________________________________________________________________________________

Rubi [A]  time = 1.21572, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}}}{2 \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}}}\right )}{2 \sqrt{7} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2 \sqrt{62-35 \sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{13 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{1}{169} \left (553+304 \sqrt{3}\right );-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

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]*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[62 - 35*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]])/(13*3^(1/4)*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]*Sqr
t[-1 + x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 92.3652, size = 374, normalized size = 0.99 \[ - \frac{\sqrt{7} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{7} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{6 \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{14 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 4\right ) \sqrt{x^{3} - 1}} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (-4 + \sqrt{3}\right )^{2}}{\left (\sqrt{3} + 4\right )^{2}}; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 4\right ) \left (\sqrt{3} + 4\right ) \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(7)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*(-x + 1)*atan(3**(3/4)*sqrt(
7)*sqrt(sqrt(3) + 2)*sqrt(-(-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3))**2 + 1)/(6*sq
rt((-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3))**2 + 4*sqrt(3) + 7)))/(14*sqrt((x - 1
)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1)) - 2*3**(3/4)*sqrt((x**2 + x + 1)/(-x -
sqrt(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)*(-sq
rt(3) + 4)*sqrt(x**3 - 1)) - 4*3**(1/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**
2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_pi((-4 + sqrt(3))**2/(sqrt(3) + 4)**2, as
in((-x + 1 + sqrt(3))/(x - 1 + sqrt(3))), -7 + 4*sqrt(3))/(sqrt((x - 1)/(-x - sq
rt(3) + 1)**2)*(-sqrt(3) + 4)*(sqrt(3) + 4)*sqrt(4*sqrt(3) + 7)*sqrt(x**3 - 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0832225, size = 126, normalized size = 0.34 \[ -\frac{4 \sqrt{2} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )}{\left (\sqrt{3}+5 i\right ) \sqrt{x^3-1}} \]

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*
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])

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 124, normalized size = 0.3 \[{\frac{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}{2}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{3}{8}}+{\frac{i}{8}}\sqrt{3},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3+x)/(x^3-1)^(1/2),x)

[Out]

1/2*(-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),3/8+1/8*I*3^(1/2),
((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)