3.694 \(\int \frac{\sqrt{-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx\)

Optimal. Leaf size=110 \[ \frac{2 \left (1-21 x^2\right )}{27 \left (x^2\right )^{3/2}}+\frac{\left (x^2-1\right ) \sec ^{-1}(x)^2}{3 \left (x^2\right )^{3/2}}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}-\frac{4 \sqrt{x^2-1} \sec ^{-1}(x)}{3 x}+\frac{\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^3}{3 x^3}-\frac{2 \left (x^2-1\right )^{3/2} \sec ^{-1}(x)}{9 x^3} \]

[Out]

(2*(1 - 21*x^2))/(27*(x^2)^(3/2)) - (4*Sqrt[-1 + x^2]*ArcSec[x])/(3*x) - (2*(-1
+ x^2)^(3/2)*ArcSec[x])/(9*x^3) + (2*ArcSec[x]^2)/(3*Sqrt[x^2]) + ((-1 + x^2)*Ar
cSec[x]^2)/(3*(x^2)^(3/2)) + ((-1 + x^2)^(3/2)*ArcSec[x]^3)/(3*x^3)

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Rubi [A]  time = 0.312982, antiderivative size = 146, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{14}{9 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}+\frac{\left (1-\frac{1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}-\frac{2 \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)}{9 x}-\frac{4 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \sec ^{-1}(x)}{3 x}+\frac{2 \sqrt{x^2}}{27 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + x^2]*ArcSec[x]^3)/x^4,x]

[Out]

-14/(9*Sqrt[x^2]) + (2*Sqrt[x^2])/(27*x^4) - (4*Sqrt[1 - x^(-2)]*Sqrt[x^2]*ArcSe
c[x])/(3*x) - (2*(1 - x^(-2))^(3/2)*Sqrt[x^2]*ArcSec[x])/(9*x) + (2*ArcSec[x]^2)
/(3*Sqrt[x^2]) + ((1 - x^(-2))*ArcSec[x]^2)/(3*Sqrt[x^2]) + ((1 - x^(-2))^(3/2)*
Sqrt[x^2]*ArcSec[x]^3)/(3*x)

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Rubi in Sympy [A]  time = 14.2463, size = 153, normalized size = 1.39 \[ \frac{\left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}} \sqrt{x^{2}} \operatorname{acos}^{3}{\left (\frac{1}{x} \right )}}{3 x} - \frac{2 \left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}} \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{9 x} - \frac{4 \sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{3 x} + \frac{\left (1 - \frac{1}{x^{2}}\right ) \sqrt{x^{2}} \operatorname{acos}^{2}{\left (\frac{1}{x} \right )}}{3 x^{2}} + \frac{2 \sqrt{x^{2}} \operatorname{acos}^{2}{\left (\frac{1}{x} \right )}}{3 x^{2}} - \frac{14 \sqrt{x^{2}}}{9 x^{2}} + \frac{2 \sqrt{x^{2}}}{27 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asec(x)**3*(x**2-1)**(1/2)/x**4,x)

[Out]

(1 - 1/x**2)**(3/2)*sqrt(x**2)*acos(1/x)**3/(3*x) - 2*(1 - 1/x**2)**(3/2)*sqrt(x
**2)*acos(1/x)/(9*x) - 4*sqrt(1 - 1/x**2)*sqrt(x**2)*acos(1/x)/(3*x) + (1 - 1/x*
*2)*sqrt(x**2)*acos(1/x)**2/(3*x**2) + 2*sqrt(x**2)*acos(1/x)**2/(3*x**2) - 14*s
qrt(x**2)/(9*x**2) + 2*sqrt(x**2)/(27*x**4)

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Mathematica [A]  time = 0.0892186, size = 92, normalized size = 0.84 \[ \frac{2 \sqrt{1-\frac{1}{x^2}} x \left (1-21 x^2\right )+9 \left (x^2-1\right )^2 \sec ^{-1}(x)^3+9 \sqrt{1-\frac{1}{x^2}} x \left (3 x^2-1\right ) \sec ^{-1}(x)^2-6 \left (7 x^4-8 x^2+1\right ) \sec ^{-1}(x)}{27 x^3 \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + x^2]*ArcSec[x]^3)/x^4,x]

[Out]

(2*Sqrt[1 - x^(-2)]*x*(1 - 21*x^2) - 6*(1 - 8*x^2 + 7*x^4)*ArcSec[x] + 9*Sqrt[1
- x^(-2)]*x*(-1 + 3*x^2)*ArcSec[x]^2 + 9*(-1 + x^2)^2*ArcSec[x]^3)/(27*x^3*Sqrt[
-1 + x^2])

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Maple [C]  time = 0.48, size = 250, normalized size = 2.3 \[{\frac{9\,i \left ({\rm arcsec} \left (x\right ) \right ) ^{2}+9\, \left ({\rm arcsec} \left (x\right ) \right ) ^{3}-2\,i-6\,{\rm arcsec} \left (x\right )}{216\,{x}^{3}} \left ( -3\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}+{x}^{4}+4\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-5\,{x}^{2}+4 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{ \left ({\rm arcsec} \left (x\right ) \right ) ^{3}-6\,{\rm arcsec} \left (x\right )+3\,i \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-6\,i}{8\,x} \left ( -i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{ \left ({\rm arcsec} \left (x\right ) \right ) ^{3}-6\,{\rm arcsec} \left (x\right )-3\,i \left ({\rm arcsec} \left (x\right ) \right ) ^{2}+6\,i}{8\,x} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{-9\,i \left ({\rm arcsec} \left (x\right ) \right ) ^{2}+9\, \left ({\rm arcsec} \left (x\right ) \right ) ^{3}+2\,i-6\,{\rm arcsec} \left (x\right )}{216\,{x}^{3}} \left ( -5\,{x}^{2}+4+3\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}+{x}^{4}-4\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsec(x)^3*(x^2-1)^(1/2)/x^4,x)

[Out]

1/216/(x^2-1)^(1/2)/x^3*(-3*I*((x^2-1)/x^2)^(1/2)*x^3+x^4+4*I*((x^2-1)/x^2)^(1/2
)*x-5*x^2+4)*(9*I*arcsec(x)^2+9*arcsec(x)^3-2*I-6*arcsec(x))+1/8/(x^2-1)^(1/2)/x
*(-I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(arcsec(x)^3-6*arcsec(x)+3*I*arcsec(x)^2-6*I)+
1/8/(x^2-1)^(1/2)*(I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(arcsec(x)^3-6*arcsec(x)-3*I*a
rcsec(x)^2+6*I)/x+1/216/(x^2-1)^(1/2)/x^3*(-5*x^2+4+3*I*((x^2-1)/x^2)^(1/2)*x^3+
x^4-4*I*((x^2-1)/x^2)^(1/2)*x)*(-9*I*arcsec(x)^2+9*arcsec(x)^3+2*I-6*arcsec(x))

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Maxima [A]  time = 1.67185, size = 100, normalized size = 0.91 \[ \frac{{\left (x^{2} - 1\right )}^{\frac{3}{2}} \operatorname{arcsec}\left (x\right )^{3}}{3 \, x^{3}} + \frac{{\left (3 \, x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )^{2}}{3 \, x^{3}} - \frac{2 \,{\left (21 \, x^{2} - 1\right )}}{27 \, x^{3}} - \frac{2 \,{\left (7 \, x^{4} - 8 \, x^{2} + 1\right )} \operatorname{arcsec}\left (x\right )}{9 \, \sqrt{x + 1} \sqrt{x - 1} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 1)*arcsec(x)^3/x^4,x, algorithm="maxima")

[Out]

1/3*(x^2 - 1)^(3/2)*arcsec(x)^3/x^3 + 1/3*(3*x^2 - 1)*arcsec(x)^2/x^3 - 2/27*(21
*x^2 - 1)/x^3 - 2/9*(7*x^4 - 8*x^2 + 1)*arcsec(x)/(sqrt(x + 1)*sqrt(x - 1)*x^3)

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Fricas [A]  time = 0.245998, size = 77, normalized size = 0.7 \[ \frac{9 \,{\left (3 \, x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )^{2} - 42 \, x^{2} + 3 \,{\left (3 \,{\left (x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )^{3} - 2 \,{\left (7 \, x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )\right )} \sqrt{x^{2} - 1} + 2}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 1)*arcsec(x)^3/x^4,x, algorithm="fricas")

[Out]

1/27*(9*(3*x^2 - 1)*arcsec(x)^2 - 42*x^2 + 3*(3*(x^2 - 1)*arcsec(x)^3 - 2*(7*x^2
- 1)*arcsec(x))*sqrt(x^2 - 1) + 2)/x^3

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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)**3*(x**2-1)**(1/2)/x**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right )^{3}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 1)*arcsec(x)^3/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(x^2 - 1)*arcsec(x)^3/x^4, x)