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

Optimal. Leaf size=65 \[ \frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{2 x \sec ^{-1}(x)}{3 \sqrt{x^2-1}}-\frac{x \sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}}+\frac{5}{6} \coth ^{-1}\left (\sqrt{x^2}\right ) \]

[Out]

Sqrt[x^2]/(6*(1 - x^2)) + (5*ArcCoth[Sqrt[x^2]])/6 - (x*ArcSec[x])/(3*(-1 + x^2)
^(3/2)) + (2*x*ArcSec[x])/(3*Sqrt[-1 + x^2])

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Rubi [A]  time = 0.0576145, antiderivative size = 67, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{2 x \sec ^{-1}(x)}{3 \sqrt{x^2-1}}-\frac{x \sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}}+\frac{5 x \tanh ^{-1}(x)}{6 \sqrt{x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[ArcSec[x]/(-1 + x^2)^(5/2),x]

[Out]

Sqrt[x^2]/(6*(1 - x^2)) - (x*ArcSec[x])/(3*(-1 + x^2)^(3/2)) + (2*x*ArcSec[x])/(
3*Sqrt[-1 + x^2]) + (5*x*ArcTanh[x])/(6*Sqrt[x^2])

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Rubi in Sympy [A]  time = 4.48212, size = 63, normalized size = 0.97 \[ \frac{x^{2}}{6 \left (- x^{2} + 1\right ) \sqrt{x^{2}}} + \frac{5 x \operatorname{atanh}{\left (x \right )}}{6 \sqrt{x^{2}}} + \frac{2 x \operatorname{asec}{\left (x \right )}}{3 \sqrt{x^{2} - 1}} - \frac{x \operatorname{asec}{\left (x \right )}}{3 \left (x^{2} - 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asec(x)/(x**2-1)**(5/2),x)

[Out]

x**2/(6*(-x**2 + 1)*sqrt(x**2)) + 5*x*atanh(x)/(6*sqrt(x**2)) + 2*x*asec(x)/(3*s
qrt(x**2 - 1)) - x*asec(x)/(3*(x**2 - 1)**(3/2))

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Mathematica [A]  time = 0.154264, size = 67, normalized size = 1.03 \[ \frac{\sqrt{1-\frac{1}{x^2}} x \left (-5 \left (x^2-1\right ) \log (1-x)+5 \left (x^2-1\right ) \log (x+1)-2 x\right )+4 x \left (2 x^2-3\right ) \sec ^{-1}(x)}{12 \left (x^2-1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSec[x]/(-1 + x^2)^(5/2),x]

[Out]

(4*x*(-3 + 2*x^2)*ArcSec[x] + Sqrt[1 - x^(-2)]*x*(-2*x - 5*(-1 + x^2)*Log[1 - x]
+ 5*(-1 + x^2)*Log[1 + x]))/(12*(-1 + x^2)^(3/2))

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Maple [C]  time = 0.414, size = 128, normalized size = 2. \[{\frac{x}{6\,{x}^{4}-12\,{x}^{2}+6}\sqrt{{x}^{2}-1} \left ( 4\,{\rm arcsec} \left (x\right ){x}^{2}-\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-6\,{\rm arcsec} \left (x\right ) \right ) }-{\frac{5\,x}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ({x}^{-1}+i\sqrt{1-{x}^{-2}}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{5\,x}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ( 1+{x}^{-1}+i\sqrt{1-{x}^{-2}} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsec(x)/(x^2-1)^(5/2),x)

[Out]

1/6*(x^2-1)^(1/2)*x/(x^4-2*x^2+1)*(4*arcsec(x)*x^2-((x^2-1)/x^2)^(1/2)*x-6*arcse
c(x))-5/6/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*ln(1/x+I*(1-1/x^2)^(1/2)-1)+5/6/(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 = 1.50952, size = 65, normalized size = 1. \[ \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{x^{2} - 1}} - \frac{x}{{\left (x^{2} - 1\right )}^{\frac{3}{2}}}\right )} \operatorname{arcsec}\left (x\right ) - \frac{x}{6 \,{\left (x^{2} - 1\right )}} + \frac{5}{12} \, \log \left (x + 1\right ) - \frac{5}{12} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="maxima")

[Out]

1/3*(2*x/sqrt(x^2 - 1) - x/(x^2 - 1)^(3/2))*arcsec(x) - 1/6*x/(x^2 - 1) + 5/12*l
og(x + 1) - 5/12*log(x - 1)

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Fricas [A]  time = 0.228902, size = 101, normalized size = 1.55 \[ -\frac{2 \, x^{3} - 4 \,{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - 5 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right ) + 5 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x - 1\right ) - 2 \, x}{12 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="fricas")

[Out]

-1/12*(2*x^3 - 4*(2*x^3 - 3*x)*sqrt(x^2 - 1)*arcsec(x) - 5*(x^4 - 2*x^2 + 1)*log
(x + 1) + 5*(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(asec(x)/(x**2-1)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259447, size = 78, normalized size = 1.2 \[ \frac{{\left (2 \, x^{2} - 3\right )} x \arccos \left (\frac{1}{x}\right )}{3 \,{\left (x^{2} - 1\right )}^{\frac{3}{2}}} + \frac{5 \,{\rm ln}\left ({\left | x + 1 \right |}\right )}{12 \,{\rm sign}\left (x\right )} - \frac{5 \,{\rm ln}\left ({\left | x - 1 \right |}\right )}{12 \,{\rm sign}\left (x\right )} - \frac{x}{6 \,{\left (x^{2} - 1\right )}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="giac")

[Out]

1/3*(2*x^2 - 3)*x*arccos(1/x)/(x^2 - 1)^(3/2) + 5/12*ln(abs(x + 1))/sign(x) - 5/
12*ln(abs(x - 1))/sign(x) - 1/6*x/((x^2 - 1)*sign(x))