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

Optimal. Leaf size=327 \[ \frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c-\sqrt{3} d+d\right )}-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (c+\sqrt{3} d+d\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}} \]

[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 + Sqr
t[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 = 1.85727, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c-\sqrt{3} d+d\right )}-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (c+\sqrt{3} d+d\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}} \]

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

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]

Timed out

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Mathematica [C]  time = 1.20336, size = 233, normalized size = 0.71 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (-\frac{3 \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt{x^2+x+1} \left (\sqrt{3} c+\left (3+\sqrt{3}\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt{x^3-1}} \]

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))]*((-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 = 0.055, size = 273, normalized size = 0.8 \[ 2\,{\frac{ \left ( c+d+d\sqrt{3} \right ) \left ( -3/2-i/2\sqrt{3} \right ) }{{d}^{2}\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},{(3/2+i/2\sqrt{3}) \left ( 1+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) \left ( 1+{\frac{c}{d}} \right ) ^{-1}}-2\,{\frac{-3/2-i/2\sqrt{3}}{d\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

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*(c+d+d*3^(1/2))/d^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)/(1+c/d)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2))
)^(1/2),(3/2+1/2*I*3^(1/2))/(1+c/d),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1
/2))-2/d*(-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)*EllipticF(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),((3/2+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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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]

Exception raised: TypeError

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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*s
qrt(x**3 - 1) + d*x*sqrt(x**3 - 1)), x) - Integral(-1/(c*sqrt(x**3 - 1) + d*x*sq
rt(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)