Optimal. Leaf size=51 \[ \frac{\sqrt{x^2}}{6 \left (1-x^2\right )}-\frac{1}{6} \coth ^{-1}\left (\sqrt{x^2}\right )-\frac{x^3 \sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.101729, antiderivative size = 53, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{x^2}}{6 \left (1-x^2\right )}-\frac{x \tanh ^{-1}(x)}{6 \sqrt{x^2}}-\frac{x^3 \sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In] Int[(x^2*ArcSec[x])/(-1 + x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 5.38584, size = 46, normalized size = 0.9 \[ - \frac{x^{3} \operatorname{asec}{\left (x \right )}}{3 \left (x^{2} - 1\right )^{\frac{3}{2}}} + \frac{x^{2}}{6 \left (- x^{2} + 1\right ) \sqrt{x^{2}}} - \frac{x \operatorname{atanh}{\left (x \right )}}{6 \sqrt{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*asec(x)/(x**2-1)**(5/2),x)
[Out]
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Mathematica [A] time = 0.139068, size = 61, normalized size = 1.2 \[ \frac{\sqrt{1-\frac{1}{x^2}} x \left (\left (x^2-1\right ) \log (1-x)-\left (x^2-1\right ) \log (x+1)-2 x\right )-4 x^3 \sec ^{-1}(x)}{12 \left (x^2-1\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*ArcSec[x])/(-1 + x^2)^(5/2),x]
[Out]
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Maple [C] time = 0.442, size = 121, normalized size = 2.4 \[ -{\frac{{x}^{2}}{6\,{x}^{4}-12\,{x}^{2}+6}\sqrt{{x}^{2}-1} \left ( 2\,x{\rm arcsec} \left (x\right )+\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}} \right ) }+{\frac{x}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ({x}^{-1}+i\sqrt{1-{x}^{-2}}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{x}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ( 1+{x}^{-1}+i\sqrt{1-{x}^{-2}} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*arcsec(x)/(x^2-1)^(5/2),x)
[Out]
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Maxima [A] time = 1.46852, size = 62, normalized size = 1.22 \[ -\frac{1}{3} \,{\left (\frac{x}{\sqrt{x^{2} - 1}} + \frac{x}{{\left (x^{2} - 1\right )}^{\frac{3}{2}}}\right )} \operatorname{arcsec}\left (x\right ) - \frac{x}{6 \,{\left (x^{2} - 1\right )}} - \frac{1}{12} \, \log \left (x + 1\right ) + \frac{1}{12} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226634, size = 92, normalized size = 1.8 \[ -\frac{4 \, \sqrt{x^{2} - 1} x^{3} \operatorname{arcsec}\left (x\right ) + 2 \, x^{3} +{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right ) -{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x - 1\right ) - 2 \, x}{12 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*arcsec(x)/(x^2 - 1)^(5/2),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*asec(x)/(x**2-1)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.256302, size = 72, normalized size = 1.41 \[ -\frac{x^{3} \arccos \left (\frac{1}{x}\right )}{3 \,{\left (x^{2} - 1\right )}^{\frac{3}{2}}} - \frac{{\rm ln}\left ({\left | x + 1 \right |}\right )}{12 \,{\rm sign}\left (x\right )} + \frac{{\rm ln}\left ({\left | x - 1 \right |}\right )}{12 \,{\rm sign}\left (x\right )} - \frac{x}{6 \,{\left (x^{2} - 1\right )}{\rm sign}\left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="giac")
[Out]