Optimal. Leaf size=136 \[ -\frac{x^2}{6}-\frac{1}{6} \sqrt{x^2+1} x+\frac{8 \sqrt{x^2+1}}{9}-\frac{7}{54} \log \left (3 x^2+2 x+3\right )+\frac{4}{27} \sqrt{2} \tan ^{-1}\left (\frac{x+1}{\sqrt{2} \sqrt{x^2+1}}\right )+\frac{7}{27} \tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+1}}\right )+\frac{8 x}{9}+\frac{4}{27} \sqrt{2} \tan ^{-1}\left (\frac{3 x+1}{2 \sqrt{2}}\right )-\frac{41}{54} \sinh ^{-1}(x) \]
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Rubi [A] time = 2.80218, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 14, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{x^2}{6}-\frac{1}{6} \sqrt{x^2+1} x+\frac{8 \sqrt{x^2+1}}{9}-\frac{7}{54} \log \left (3 x^2+2 x+3\right )+\frac{4}{27} \sqrt{2} \tan ^{-1}\left (\frac{x+1}{\sqrt{2} \sqrt{x^2+1}}\right )+\frac{7}{27} \tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+1}}\right )+\frac{8 x}{9}+\frac{4}{27} \sqrt{2} \tan ^{-1}\left (\frac{3 x+1}{2 \sqrt{2}}\right )-\frac{41}{54} \sinh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(x^2*(2 - Sqrt[1 + x^2]))/(Sqrt[1 + x^2]*(1 - x^3 + (1 + x^2)^(3/2))),x]
[Out]
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Rubi in Sympy [A] time = 92.1603, size = 112, normalized size = 0.82 \[ \frac{8 x}{9} - \frac{\left (x + \sqrt{x^{2} + 1}\right )^{2}}{12} + \frac{8 \sqrt{x^{2} + 1}}{9} - \frac{\log{\left (x + \sqrt{x^{2} + 1} \right )}}{2} - \frac{7 \log{\left (- 2 x + 3 \left (x + \sqrt{x^{2} + 1}\right )^{2} - 2 \sqrt{x^{2} + 1} + 1 \right )}}{27} + \frac{8 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 x}{2} + \frac{3 \sqrt{x^{2} + 1}}{2} - \frac{1}{2}\right ) \right )}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(2-(x**2+1)**(1/2))/(1-x**3+(x**2+1)**(3/2))/(x**2+1)**(1/2),x)
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Mathematica [C] time = 6.34784, size = 1046, normalized size = 7.69 \[ -\frac{x^2}{6}+\frac{8 x}{9}-\frac{41}{54} \sinh ^{-1}(x)+\frac{4}{27} \sqrt{2} \tan ^{-1}\left (\frac{3 x+1}{2 \sqrt{2}}\right )+\frac{\left (i+11 \sqrt{2}\right ) \tan ^{-1}\left (\frac{3447 i \sqrt{2} x^4-792 x^4-2187 i \sqrt{2 \left (-1+2 i \sqrt{2}\right )} \sqrt{x^2+1} x^3+2760 i \sqrt{2} x^3+11040 x^3+5103 i \sqrt{2 \left (-1+2 i \sqrt{2}\right )} \sqrt{x^2+1} x^2-862 i \sqrt{2} x^2+3680 x^2+2187 i \sqrt{2 \left (-1+2 i \sqrt{2}\right )} \sqrt{x^2+1} x+2760 i \sqrt{2} x+11040 x+6561 i \sqrt{2 \left (-1+2 i \sqrt{2}\right )} \sqrt{x^2+1}-3913 i \sqrt{2}+4472}{8712 \sqrt{2} x^4+22581 i x^4+6864 \sqrt{2} x^3+3432 i x^3+10064 \sqrt{2} x^2+39862 i x^2+6864 \sqrt{2} x+3432 i x+1352 \sqrt{2}+17317 i}\right )}{54 \sqrt{-1+2 i \sqrt{2}}}-\frac{i \left (-i+11 \sqrt{2}\right ) \tan ^{-1}\left (\frac{2 \left (4356 \sqrt{2} x^4-1449 i x^4+3432 \sqrt{2} x^3-1716 i x^3+5032 \sqrt{2} x^2-4622 i x^2+3432 \sqrt{2} x-1716 i x+676 \sqrt{2}+1183 i\right )}{3447 i \sqrt{2} x^4+792 x^4+8748 i \sqrt{1+2 i \sqrt{2}} \sqrt{x^2+1} x^3+2760 i \sqrt{2} x^3-11040 x^3+5832 i \sqrt{1+2 i \sqrt{2}} \sqrt{x^2+1} x^2-862 i \sqrt{2} x^2-3680 x^2+8748 i \sqrt{1+2 i \sqrt{2}} \sqrt{x^2+1} x+2760 i \sqrt{2} x-11040 x-3913 i \sqrt{2}-4472}\right )}{54 \sqrt{1+2 i \sqrt{2}}}+\frac{i \left (i+11 \sqrt{2}\right ) \log \left (\left (-3 i x+2 \sqrt{2}-i\right )^2 \left (3 i x+2 \sqrt{2}+i\right )^2\right )}{108 \sqrt{-1+2 i \sqrt{2}}}-\frac{\left (-i+11 \sqrt{2}\right ) \log \left (\left (-3 i x+2 \sqrt{2}-i\right )^2 \left (3 i x+2 \sqrt{2}+i\right )^2\right )}{108 \sqrt{1+2 i \sqrt{2}}}-\frac{7}{54} \log \left (3 x^2+2 x+3\right )-\frac{i \left (i+11 \sqrt{2}\right ) \log \left (\left (3 x^2+2 x+3\right ) \left (4 \sqrt{2} x^2-7 i x^2-8 i \sqrt{-1+2 i \sqrt{2}} \sqrt{x^2+1} x+6 i x+4 \sqrt{2}-7 i\right )\right )}{108 \sqrt{-1+2 i \sqrt{2}}}+\frac{\left (-i+11 \sqrt{2}\right ) \log \left (\left (3 x^2+2 x+3\right ) \left (4 \sqrt{2} x^2-11 i x^2+2 i \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{x^2+1} x+6 i x-6 i \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{x^2+1}+4 \sqrt{2}-11 i\right )\right )}{108 \sqrt{1+2 i \sqrt{2}}}+\left (\frac{8}{9}-\frac{x}{6}\right ) \sqrt{x^2+1} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(2 - Sqrt[1 + x^2]))/(Sqrt[1 + x^2]*(1 - x^3 + (1 + x^2)^(3/2))),x]
[Out]
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Maple [B] time = 0.079, size = 654, normalized size = 4.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(2-(x^2+1)^(1/2))/(1-x^3+(x^2+1)^(3/2))/(x^2+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) + \int -\frac{3 \, x^{10} - 4 \, x^{9} + 5 \, x^{8} - 2 \, x^{7} + 15 \, x^{6} + 6 \, x^{5} + 9 \, x^{4}}{2 \,{\left (2 \, x^{13} + 7 \, x^{11} - 4 \, x^{10} + 11 \, x^{9} - 11 \, x^{8} + 13 \, x^{7} - 13 \, x^{6} + 11 \, x^{5} - 11 \, x^{4} + 4 \, x^{3} - 7 \, x^{2} - 2 \,{\left (x^{12} + 3 \, x^{10} - 2 \, x^{9} + 3 \, x^{8} - 6 \, x^{7} + 2 \, x^{6} - 6 \, x^{5} + 3 \, x^{4} - 2 \, x^{3} + 3 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} - 2\right )}}\,{d x} + \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*(sqrt(x^2 + 1) - 2)/((x^3 - (x^2 + 1)^(3/2) - 1)*sqrt(x^2 + 1)),x, algorithm="maxima")
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Fricas [A] time = 0.241463, size = 439, normalized size = 3.23 \[ \frac{8 \, \sqrt{2}{\left (2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) + 9 \, x^{2} - 8 \,{\left (2 \, \sqrt{2} \sqrt{x^{2} + 1} x - \sqrt{2}{\left (2 \, x^{2} + 1\right )}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 3 \, \sqrt{x^{2} + 1} - 1\right )}\right ) + 8 \,{\left (2 \, \sqrt{2} \sqrt{x^{2} + 1} x - \sqrt{2}{\left (2 \, x^{2} + 1\right )}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (x - \sqrt{x^{2} + 1} + 1\right )}\right ) + 7 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )} \log \left (3 \, x^{2} - \sqrt{x^{2} + 1}{\left (3 \, x - 1\right )} - x + 2\right ) - 7 \,{\left (2 \, x^{2} + 1\right )} \log \left (3 \, x^{2} + 2 \, x + 3\right ) - 7 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )} \log \left (x^{2} - \sqrt{x^{2} + 1}{\left (x + 1\right )} + x + 2\right ) + 41 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) -{\left (16 \, \sqrt{2} x \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - 14 \, x \log \left (3 \, x^{2} + 2 \, x + 3\right ) + 9 \, x - 48\right )} \sqrt{x^{2} + 1} - 48 \, x}{54 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*(sqrt(x^2 + 1) - 2)/((x^3 - (x^2 + 1)^(3/2) - 1)*sqrt(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [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**2*(2-(x**2+1)**(1/2))/(1-x**3+(x**2+1)**(3/2))/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.22043, size = 238, normalized size = 1.75 \[ -\frac{1}{6} \, x^{2} - \frac{1}{18} \, \sqrt{x^{2} + 1}{\left (3 \, x - 16\right )} + \frac{4}{27} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 3 \, \sqrt{x^{2} + 1} - 1\right )}\right ) + \frac{4}{27} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{4}{27} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (x - \sqrt{x^{2} + 1} + 1\right )}\right ) + \frac{8}{9} \, x + \frac{7}{54} \,{\rm ln}\left (3 \,{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 2 \, x + 2 \, \sqrt{x^{2} + 1} + 1\right ) - \frac{7}{54} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 1}\right )}^{2} + 2 \, x - 2 \, \sqrt{x^{2} + 1} + 3\right ) - \frac{7}{54} \,{\rm ln}\left (3 \, x^{2} + 2 \, x + 3\right ) + \frac{41}{54} \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*(sqrt(x^2 + 1) - 2)/((x^3 - (x^2 + 1)^(3/2) - 1)*sqrt(x^2 + 1)),x, algorithm="giac")
[Out]