3.687 \(\int \frac{x^2 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx\)

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]

Sqrt[x^2]/(6*(1 - x^2)) - ArcCoth[Sqrt[x^2]]/6 - (x^3*ArcSec[x])/(3*(-1 + x^2)^(
3/2))

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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]

Sqrt[x^2]/(6*(1 - x^2)) - (x^3*ArcSec[x])/(3*(-1 + x^2)^(3/2)) - (x*ArcTanh[x])/
(6*Sqrt[x^2])

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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]

-x**3*asec(x)/(3*(x**2 - 1)**(3/2)) + x**2/(6*(-x**2 + 1)*sqrt(x**2)) - x*atanh(
x)/(6*sqrt(x**2))

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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]

(-4*x^3*ArcSec[x] + Sqrt[1 - x^(-2)]*x*(-2*x + (-1 + x^2)*Log[1 - x] - (-1 + x^2
)*Log[1 + x]))/(12*(-1 + x^2)^(3/2))

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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]

-1/6*(x^2-1)^(1/2)*x^2/(x^4-2*x^2+1)*(2*x*arcsec(x)+((x^2-1)/x^2)^(1/2))+1/6/(x^
2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*ln(1/x+I*(1-1/x^2)^(1/2)-1)-1/6/(x^2-1)^(1/2)*(
(x^2-1)/x^2)^(1/2)*x*ln(1+1/x+I*(1-1/x^2)^(1/2))

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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]

-1/3*(x/sqrt(x^2 - 1) + x/(x^2 - 1)^(3/2))*arcsec(x) - 1/6*x/(x^2 - 1) - 1/12*lo
g(x + 1) + 1/12*log(x - 1)

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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]

-1/12*(4*sqrt(x^2 - 1)*x^3*arcsec(x) + 2*x^3 + (x^4 - 2*x^2 + 1)*log(x + 1) - (x
^4 - 2*x^2 + 1)*log(x - 1) - 2*x)/(x^4 - 2*x^2 + 1)

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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]

Timed 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]

-1/3*x^3*arccos(1/x)/(x^2 - 1)^(3/2) - 1/12*ln(abs(x + 1))/sign(x) + 1/12*ln(abs
(x - 1))/sign(x) - 1/6*x/((x^2 - 1)*sign(x))