Optimal. Leaf size=344 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (c-\sqrt{3} d+d\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}-\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\sqrt{3} d+d\right )^2}{\left (c+\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c+\sqrt{3} d+d\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.8248, antiderivative size = 344, 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 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (c-\sqrt{3} d+d\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}-\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\sqrt{3} d+d\right )^2}{\left (c+\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c+\sqrt{3} d+d\right )} \]
Antiderivative was successfully verified.
[In] Int[(1 - Sqrt[3] - x)/((c + d*x)*Sqrt[-1 + x^3]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 154.739, size = 325, normalized size = 0.94 \[ - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (c - \sqrt{3} d + d\right )^{2}}{\left (c + d + \sqrt{3} d\right )^{2}}; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1} \left (c + d + \sqrt{3} d\right )} - \frac{\sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) \left (c - \sqrt{3} d + d\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1} \sqrt{c^{2} - c d + d^{2}}}{3 \sqrt{d} \sqrt{c + d} \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{\sqrt{d} \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{c + d} \sqrt{x^{3} - 1} \sqrt{c^{2} - c d + d^{2}}} \]
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]
_______________________________________________________________________________________
Mathematica [C] time = 1.17288, size = 233, normalized size = 0.68 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (-\frac{3 \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt{x^2+x+1} \left (\sqrt{3} c+\left (\sqrt{3}-3\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt{x^3-1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 - Sqrt[3] - x)/((c + d*x)*Sqrt[-1 + x^3]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.028, size = 277, normalized size = 0.8 \[ -2\,{\frac{-3/2-i/2\sqrt{3}}{d\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-2\,{\frac{ \left ( d\sqrt{3}-c-d \right ) \left ( -3/2-i/2\sqrt{3} \right ) }{{d}^{2}\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},{(3/2+i/2\sqrt{3}) \left ( 1+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) \left ( 1+{\frac{c}{d}} \right ) ^{-1}} \]
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{3}}{c \sqrt{x^{3} - 1} + d x \sqrt{x^{3} - 1}}\, dx - \int \frac{x}{c \sqrt{x^{3} - 1} + d x \sqrt{x^{3} - 1}}\, dx - \int \left (- \frac{1}{c \sqrt{x^{3} - 1} + d x \sqrt{x^{3} - 1}}\right )\, dx \]
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]
_______________________________________________________________________________________
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]