Optimal. Leaf size=543 \[ \frac{\left (1-x^2\right )^{2/3} x}{24 \left (x^2+3\right )}-\frac{x}{24 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}+\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.594902, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{\left (1-x^2\right )^{2/3} x}{24 \left (x^2+3\right )}-\frac{x}{24 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}+\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}} \]
Warning: Unable to verify antiderivative.
[In] Int[1/((1 - x^2)^(1/3)*(3 + x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 39.6816, size = 456, normalized size = 0.84 \[ \frac{x \left (- x^{2} + 1\right )^{\frac{2}{3}}}{24 \left (x^{2} + 3\right )} - \frac{x}{24 \left (- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1\right )} + \frac{\sqrt [3]{2} \log{\left (\sqrt [3]{2} \sqrt [3]{- x + 1} + \left (x + 1\right )^{\frac{2}{3}} \right )}}{32} - \frac{\sqrt [3]{2} \log{\left (\left (- x + 1\right )^{\frac{2}{3}} + \sqrt [3]{2} \sqrt [3]{x + 1} \right )}}{32} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{- x + 1}} \right )}}{48} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (- x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{x + 1}} - \frac{\sqrt{3}}{3} \right )}}{48} - \frac{\sqrt [4]{3} \sqrt{\frac{\left (- x^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- x^{2} + 1} + 1}{\left (- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{- x^{2} + 1} + 1\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- x^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{48 x \sqrt{\frac{\sqrt [3]{- x^{2} + 1} - 1}{\left (- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1\right )^{2}}}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{\left (- x^{2} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- x^{2} + 1} + 1}{\left (- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt [3]{- x^{2} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- x^{2} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{72 x \sqrt{\frac{\sqrt [3]{- x^{2} + 1} - 1}{\left (- \sqrt [3]{- x^{2} + 1} - \sqrt{3} + 1\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2+1)**(1/3)/(x**2+3)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.205081, size = 231, normalized size = 0.43 \[ \frac{x \left (\frac{5 x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )+15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}+\frac{189 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}-3 x^2+3\right )}{72 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((1 - x^2)^(1/3)*(3 + x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2+1)^(1/3)/(x^2+3)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
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(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2+1)**(1/3)/(x**2+3)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}^{2}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 3)^2*(-x^2 + 1)^(1/3)),x, algorithm="giac")
[Out]