Optimal. Leaf size=67 \[ -\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sqrt{1-x^2}}{3 x^3} \]
[Out]
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Rubi [A] time = 0.167301, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sqrt{1-x^2}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[(1 + x)^2/(x^4*Sqrt[1 - x^2]),x]
[Out]
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Rubi in Sympy [A] time = 12.5477, size = 49, normalized size = 0.73 \[ - \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} - \frac{5 \sqrt{- x^{2} + 1}}{3 x} - \frac{\sqrt{- x^{2} + 1}}{x^{2}} - \frac{\sqrt{- x^{2} + 1}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)**2/x**4/(-x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0498194, size = 47, normalized size = 0.7 \[ -\log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \left (5 x^2+3 x+1\right )}{3 x^3}+\log (x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x)^2/(x^4*Sqrt[1 - x^2]),x]
[Out]
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Maple [A] time = 0.012, size = 56, normalized size = 0.8 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{5}{3\,x}\sqrt{-{x}^{2}+1}}-{\frac{1}{{x}^{2}}\sqrt{-{x}^{2}+1}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)^2/x^4/(-x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 0.792782, size = 92, normalized size = 1.37 \[ -\frac{5 \, \sqrt{-x^{2} + 1}}{3 \, x} - \frac{\sqrt{-x^{2} + 1}}{x^{2}} - \frac{\sqrt{-x^{2} + 1}}{3 \, x^{3}} - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272534, size = 194, normalized size = 2.9 \[ -\frac{5 \, x^{6} + 3 \, x^{5} - 24 \, x^{4} - 15 \, x^{3} + 15 \, x^{2} - 3 \,{\left (3 \, x^{5} - 4 \, x^{3} -{\left (x^{5} - 4 \, x^{3}\right )} \sqrt{-x^{2} + 1}\right )} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) +{\left (15 \, x^{4} + 9 \, x^{3} - 17 \, x^{2} - 12 \, x - 4\right )} \sqrt{-x^{2} + 1} + 12 \, x + 4}{3 \,{\left (3 \, x^{5} - 4 \, x^{3} -{\left (x^{5} - 4 \, x^{3}\right )} \sqrt{-x^{2} + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.2272, size = 128, normalized size = 1.91 \[ \begin{cases} - \frac{\sqrt{- x^{2} + 1}}{x} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3 x^{3}} & \text{for}\: x > -1 \wedge x < 1 \end{cases} + \begin{cases} - \frac{i \sqrt{x^{2} - 1}}{x} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{\sqrt{- x^{2} + 1}}{x} & \text{otherwise} \end{cases} + 2 \left (\begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x} \right )}}{2} - \frac{\sqrt{-1 + \frac{1}{x^{2}}}}{2 x} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x} \right )}}{2} - \frac{i}{2 x \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{2 x^{3} \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)**2/x**4/(-x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.295301, size = 169, normalized size = 2.52 \[ -\frac{x^{3}{\left (\frac{6 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{21 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{24 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}} - \frac{7 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{8 \, x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{24 \, x^{3}} +{\rm ln}\left (-\frac{\sqrt{-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^4),x, algorithm="giac")
[Out]