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

Optimal. Leaf size=364 \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1-\sqrt{3}\right ) d\right )^2}{\left (c-\left (1+\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\sqrt{3} d-d\right )}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left (x+\sqrt{3}+1\right )^2}{\left (x-\sqrt{3}+1\right )^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}} \]

[Out]

-(((c - (1 - Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*ArcTanh
[(2*Sqrt[2 + Sqrt[3]]*Sqrt[c^2 + c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]
)/(Sqrt[c - d]*Sqrt[d]*Sqrt[7 + 4*Sqrt[3] + (1 + Sqrt[3] + x)^2/(1 - Sqrt[3] + x
)^2])])/(Sqrt[c - d]*Sqrt[d]*Sqrt[c^2 + c*d + d^2]*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[(c - (1 - Sqrt[3])*d)^2/(c - (1 + Sqrt[3])*d)
^2, -ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/((c - d - Sqr
t[3]*d)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi [A]  time = 2.09602, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1-\sqrt{3}\right ) d\right )^2}{\left (c-\left (1+\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\sqrt{3} d-d\right )}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left (x+\sqrt{3}+1\right )^2}{\left (x-\sqrt{3}+1\right )^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((c - (1 - Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*ArcTanh
[(2*Sqrt[2 + Sqrt[3]]*Sqrt[c^2 + c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]
)/(Sqrt[c - d]*Sqrt[d]*Sqrt[7 + 4*Sqrt[3] + (1 + Sqrt[3] + x)^2/(1 - Sqrt[3] + x
)^2])])/(Sqrt[c - d]*Sqrt[d]*Sqrt[c^2 + c*d + d^2]*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[(c - (1 - Sqrt[3])*d)^2/(c - (1 + Sqrt[3])*d)
^2, -ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/((c - d - Sqr
t[3]*d)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi in Sympy [A]  time = 170.995, size = 330, normalized size = 0.91 \[ \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \Pi \left (\frac{\left (c - d + \sqrt{3} d\right )^{2}}{\left (- c + d + \sqrt{3} d\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \left (c - d \left (1 + \sqrt{3}\right )\right ) \sqrt{- x^{3} - 1}} - \frac{\sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \left (c - d + \sqrt{3} d\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \sqrt{c^{2} + c d + d^{2}}}{3 \sqrt{d} \sqrt{c - d} \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{d} \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{c - d} \sqrt{- x^{3} - 1} \sqrt{c^{2} + c d + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*3**(1/4)*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(x + 1)*e
lliptic_pi((c - d + sqrt(3)*d)**2/(-c + d + sqrt(3)*d)**2, asin((x + 1 + sqrt(3)
)/(-x - 1 + sqrt(3))), -7 + 4*sqrt(3))/(sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*sqrt
(4*sqrt(3) + 7)*(c - d*(1 + sqrt(3)))*sqrt(-x**3 - 1)) - sqrt((x**2 - x + 1)/(x
- sqrt(3) + 1)**2)*(x + 1)*(c - d + sqrt(3)*d)*atanh(3**(3/4)*sqrt(1 - (x + 1 +
sqrt(3))**2/(-x - 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*sqrt(c**2 + c*d + d**2)/(3*
sqrt(d)*sqrt(c - d)*sqrt(4*sqrt(3) + 7 + (x + 1 + sqrt(3))**2/(-x - 1 + sqrt(3))
**2)))/(sqrt(d)*sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*sqrt(c - d)*sqrt(-x**3 - 1)*
sqrt(c**2 + c*d + d**2))

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Mathematica [C]  time = 1.0221, size = 215, normalized size = 0.59 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left (c+\left (\sqrt{3}-1\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(
2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/
3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*(c + (-1 + S
qrt[3])*d)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin
[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(
d*Sqrt[-1 - x^3])

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Maple [A]  time = 0.029, size = 266, normalized size = 0.7 \[{\frac{{\frac{2\,i}{3}} \left ( d\sqrt{3}+c-d \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}}-{\frac{{\frac{2\,i}{3}}\sqrt{3}}{d}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x-3^(1/2))/(d*x+c)/(-x^3-1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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