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