3.693 \(\int \frac{\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx\)

Optimal. Leaf size=133 \[ \frac{x \sec ^{-1}(x)^3}{8 \sqrt{x^2}}-\frac{3 \sqrt{x^2-1} \sec ^{-1}(x)^2}{8 x^2}+\frac{9 x \sec ^{-1}(x)}{64 \sqrt{x^2}}-\frac{3 \sec ^{-1}(x)}{8 x \sqrt{x^2}}+\frac{\sqrt{x^2-1} \left (17 x^2-2\right )}{64 x^4}-\frac{\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^2}{4 x^4}+\frac{\left (x^2-1\right )^2 \sec ^{-1}(x)}{8 x^3 \sqrt{x^2}} \]

[Out]

(Sqrt[-1 + x^2]*(-2 + 17*x^2))/(64*x^4) - (3*ArcSec[x])/(8*x*Sqrt[x^2]) + (9*x*A
rcSec[x])/(64*Sqrt[x^2]) + ((-1 + x^2)^2*ArcSec[x])/(8*x^3*Sqrt[x^2]) - (3*Sqrt[
-1 + x^2]*ArcSec[x]^2)/(8*x^2) - ((-1 + x^2)^(3/2)*ArcSec[x]^2)/(4*x^4) + (x*Arc
Sec[x]^3)/(8*Sqrt[x^2])

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Rubi [A]  time = 0.308679, antiderivative size = 172, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{\left (1-\frac{1}{x^2}\right )^{3/2}}{32 \sqrt{x^2}}+\frac{15 \sqrt{1-\frac{1}{x^2}}}{64 \sqrt{x^2}}-\frac{9 \sqrt{x^2} \csc ^{-1}(x)}{64 x}+\frac{\sqrt{x^2} \sec ^{-1}(x)^3}{8 x}-\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt{x^2}}-\frac{3 \sqrt{1-\frac{1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^2 \sqrt{x^2} \sec ^{-1}(x)}{8 x}-\frac{3 \sqrt{x^2} \sec ^{-1}(x)}{8 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((-1 + x^2)^(3/2)*ArcSec[x]^2)/x^5,x]

[Out]

(15*Sqrt[1 - x^(-2)])/(64*Sqrt[x^2]) + (1 - x^(-2))^(3/2)/(32*Sqrt[x^2]) - (9*Sq
rt[x^2]*ArcCsc[x])/(64*x) - (3*Sqrt[x^2]*ArcSec[x])/(8*x^3) + ((1 - x^(-2))^2*Sq
rt[x^2]*ArcSec[x])/(8*x) - (3*Sqrt[1 - x^(-2)]*ArcSec[x]^2)/(8*Sqrt[x^2]) - ((1
- x^(-2))^(3/2)*ArcSec[x]^2)/(4*Sqrt[x^2]) + (Sqrt[x^2]*ArcSec[x]^3)/(8*x)

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Rubi in Sympy [A]  time = 14.8492, size = 180, normalized size = 1.35 \[ \frac{\left (1 - \frac{1}{x^{2}}\right )^{2} \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{8 x} + \frac{\sqrt{x^{2}} \operatorname{acos}^{3}{\left (\frac{1}{x} \right )}}{8 x} - \frac{9 \sqrt{x^{2}} \operatorname{asin}{\left (\frac{1}{x} \right )}}{64 x} - \frac{\left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}} \sqrt{x^{2}} \operatorname{acos}^{2}{\left (\frac{1}{x} \right )}}{4 x^{2}} + \frac{\left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}} \sqrt{x^{2}}}{32 x^{2}} - \frac{3 \sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2}} \operatorname{acos}^{2}{\left (\frac{1}{x} \right )}}{8 x^{2}} + \frac{15 \sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2}}}{64 x^{2}} - \frac{3 \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{8 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-1)**(3/2)*asec(x)**2/x**5,x)

[Out]

(1 - 1/x**2)**2*sqrt(x**2)*acos(1/x)/(8*x) + sqrt(x**2)*acos(1/x)**3/(8*x) - 9*s
qrt(x**2)*asin(1/x)/(64*x) - (1 - 1/x**2)**(3/2)*sqrt(x**2)*acos(1/x)**2/(4*x**2
) + (1 - 1/x**2)**(3/2)*sqrt(x**2)/(32*x**2) - 3*sqrt(1 - 1/x**2)*sqrt(x**2)*aco
s(1/x)**2/(8*x**2) + 15*sqrt(1 - 1/x**2)*sqrt(x**2)/(64*x**2) - 3*sqrt(x**2)*aco
s(1/x)/(8*x**3)

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Mathematica [A]  time = 0.324673, size = 84, normalized size = 0.63 \[ \frac{\sqrt{x^2-1} \left (32 \sec ^{-1}(x)^3+4 \sec ^{-1}(x) \left (\cos \left (4 \sec ^{-1}(x)\right )-16 \cos \left (2 \sec ^{-1}(x)\right )\right )+8 \sec ^{-1}(x)^2 \left (\sin \left (4 \sec ^{-1}(x)\right )-8 \sin \left (2 \sec ^{-1}(x)\right )\right )+32 \sin \left (2 \sec ^{-1}(x)\right )-\sin \left (4 \sec ^{-1}(x)\right )\right )}{256 \sqrt{1-\frac{1}{x^2}} x} \]

Antiderivative was successfully verified.

[In]  Integrate[((-1 + x^2)^(3/2)*ArcSec[x]^2)/x^5,x]

[Out]

(Sqrt[-1 + x^2]*(32*ArcSec[x]^3 + 4*ArcSec[x]*(-16*Cos[2*ArcSec[x]] + Cos[4*ArcS
ec[x]]) + 32*Sin[2*ArcSec[x]] - Sin[4*ArcSec[x]] + 8*ArcSec[x]^2*(-8*Sin[2*ArcSe
c[x]] + Sin[4*ArcSec[x]])))/(256*Sqrt[1 - x^(-2)]*x)

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Maple [C]  time = 0.455, size = 327, normalized size = 2.5 \[{\frac{x \left ({\rm arcsec} \left (x\right ) \right ) ^{3}}{8}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{4\,i{\rm arcsec} \left (x\right )+8\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1}{512\,{x}^{4}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{5}-8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}+4\,{x}^{4}+8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-12\,{x}^{2}+8 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{2\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1+2\,i{\rm arcsec} \left (x\right )}{16\,{x}^{2}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+2\,{x}^{2}-2 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{2\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1-2\,i{\rm arcsec} \left (x\right )}{16\,{x}^{2}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,{x}^{2}+2 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{-4\,i{\rm arcsec} \left (x\right )+8\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1}{512\,{x}^{4}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{5}-8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-4\,{x}^{4}+8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+12\,{x}^{2}-8 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-1)^(3/2)*arcsec(x)^2/x^5,x)

[Out]

1/8/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*arcsec(x)^3-1/512/(x^2-1)^(1/2)/x^4*(I*(
(x^2-1)/x^2)^(1/2)*x^5-8*I*((x^2-1)/x^2)^(1/2)*x^3+4*x^4+8*I*((x^2-1)/x^2)^(1/2)
*x-12*x^2+8)*(4*I*arcsec(x)+8*arcsec(x)^2-1)-1/16/(x^2-1)^(1/2)/x^2*(I*((x^2-1)/
x^2)^(1/2)*x^3-2*I*((x^2-1)/x^2)^(1/2)*x+2*x^2-2)*(2*arcsec(x)^2-1+2*I*arcsec(x)
)+1/16/(x^2-1)^(1/2)/x^2*(I*((x^2-1)/x^2)^(1/2)*x^3-2*I*((x^2-1)/x^2)^(1/2)*x-2*
x^2+2)*(2*arcsec(x)^2-1-2*I*arcsec(x))+1/512/(x^2-1)^(1/2)/x^4*(I*((x^2-1)/x^2)^
(1/2)*x^5-8*I*((x^2-1)/x^2)^(1/2)*x^3-4*x^4+8*I*((x^2-1)/x^2)^(1/2)*x+12*x^2-8)*
(-4*I*arcsec(x)+8*arcsec(x)^2-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.226313, size = 80, normalized size = 0.6 \[ \frac{8 \, x^{4} \operatorname{arcsec}\left (x\right )^{3} +{\left (17 \, x^{4} - 40 \, x^{2} + 8\right )} \operatorname{arcsec}\left (x\right ) -{\left (8 \,{\left (5 \, x^{2} - 2\right )} \operatorname{arcsec}\left (x\right )^{2} - 17 \, x^{2} + 2\right )} \sqrt{x^{2} - 1}}{64 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="fricas")

[Out]

1/64*(8*x^4*arcsec(x)^3 + (17*x^4 - 40*x^2 + 8)*arcsec(x) - (8*(5*x^2 - 2)*arcse
c(x)^2 - 17*x^2 + 2)*sqrt(x^2 - 1))/x^4

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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**2-1)**(3/2)*asec(x)**2/x**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} - 1\right )}^{\frac{3}{2}} \operatorname{arcsec}\left (x\right )^{2}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="giac")

[Out]

integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5, x)