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

Optimal. Leaf size=331 \[ \frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c-\sqrt{3} d+d\right )}-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (c+\sqrt{3} d+d\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}} \]

[Out]

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

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Rubi [A]  time = 2.45629, antiderivative size = 331, 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 \[ \frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c-\sqrt{3} d+d\right )}-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (c+\sqrt{3} d+d\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 165.609, size = 325, normalized size = 0.98 \[ \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (c + d + \sqrt{3} d\right )^{2}}{\left (c - \sqrt{3} d + d\right )^{2}}; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \sqrt{- x^{3} + 1} \left (c - \sqrt{3} d + d\right )} - \frac{\sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) \left (c + d + \sqrt{3} d\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt{c^{2} - c d + d^{2}}}{3 \sqrt{d} \sqrt{c + d} \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{d} \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{c + d} \sqrt{- x^{3} + 1} \sqrt{c^{2} - c d + d^{2}}} \]

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 + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1
)*elliptic_pi((c + d + sqrt(3)*d)**2/(c - sqrt(3)*d + d)**2, asin((x - 1 + sqrt(
3))/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*s
qrt(-4*sqrt(3) + 7)*sqrt(-x**3 + 1)*(c - sqrt(3)*d + d)) - sqrt((x**2 + x + 1)/(
-x + 1 + sqrt(3))**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 + sq
rt(3))**2)))/(sqrt(d)*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(c + d)*sqrt(-x**
3 + 1)*sqrt(c**2 - c*d + d**2))

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Mathematica [C]  time = 1.2329, size = 235, normalized size = 0.71 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (-\frac{3 \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt{x^2+x+1} \left (\sqrt{3} c+\left (3+\sqrt{3}\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt{1-x^3}} \]

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

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Maple [A]  time = 0.064, size = 264, normalized size = 0.8 \[{\frac{-{\frac{2\,i}{3}} \left ( c+d+d\sqrt{3} \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}} \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}}+{\frac{{\frac{2\,i}{3}}\sqrt{3}}{d}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

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

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