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

Optimal. Leaf size=70 \[ \frac{1}{2} \sqrt{x^4-1} \sec ^{-1}(x)-\frac{\sqrt{x^4-1}}{2 \sqrt{1-\frac{1}{x^2}} x}+\frac{1}{2} \tanh ^{-1}\left (\frac{\sqrt{1-\frac{1}{x^2}} x}{\sqrt{x^4-1}}\right ) \]

[Out]

-Sqrt[-1 + x^4]/(2*Sqrt[1 - x^(-2)]*x) + (Sqrt[-1 + x^4]*ArcSec[x])/2 + ArcTanh[
(Sqrt[1 - x^(-2)]*x)/Sqrt[-1 + x^4]]/2

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Rubi [A]  time = 0.333074, antiderivative size = 94, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ \frac{1}{2} \sqrt{x^4-1} \sec ^{-1}(x)-\frac{\sqrt{x^4-1}}{2 \sqrt{1-\frac{1}{x^2}} x}+\frac{\sqrt{1-x^2} \tan ^{-1}\left (\frac{\sqrt{x^4-1}}{\sqrt{1-x^2}}\right )}{2 \sqrt{1-\frac{1}{x^2}} x} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[-1 + x^4]/(2*Sqrt[1 - x^(-2)]*x) + (Sqrt[-1 + x^4]*ArcSec[x])/2 + (Sqrt[1
- x^2]*ArcTan[Sqrt[-1 + x^4]/Sqrt[1 - x^2]])/(2*Sqrt[1 - x^(-2)]*x)

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Rubi in Sympy [A]  time = 10.8354, size = 68, normalized size = 0.97 \[ \frac{x \operatorname{atanh}{\left (\frac{\sqrt{x^{4} - 1}}{\sqrt{x^{2} - 1}} \right )}}{2 \sqrt{x^{2}}} - \frac{x \sqrt{x^{4} - 1}}{2 \sqrt{x^{2} - 1} \sqrt{x^{2}}} + \frac{\sqrt{x^{4} - 1} \operatorname{asec}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*atanh(sqrt(x**4 - 1)/sqrt(x**2 - 1))/(2*sqrt(x**2)) - x*sqrt(x**4 - 1)/(2*sqrt
(x**2 - 1)*sqrt(x**2)) + sqrt(x**4 - 1)*asec(x)/2

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Mathematica [A]  time = 0.187791, size = 88, normalized size = 1.26 \[ \frac{1}{2} \left (\sqrt{x^4-1} \sec ^{-1}(x)-\log \left (x-x^3\right )-\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^4-1} x}{x^2-1}+\log \left (-x^2-\sqrt{1-\frac{1}{x^2}} \sqrt{x^4-1} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.667, size = 0, normalized size = 0. \[ \int{{x}^{3}{\rm arcsec} \left (x\right ){\frac{1}{\sqrt{{x}^{4}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.6877, size = 38, normalized size = 0.54 \[ \frac{1}{2} \, \sqrt{x^{4} - 1} \operatorname{arcsec}\left (x\right ) - \frac{1}{2} \, \sqrt{x^{2} + 1} + \frac{1}{2} \, \operatorname{arsinh}\left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^4 - 1)*arcsec(x) - 1/2*sqrt(x^2 + 1) + 1/2*arcsinh(1/abs(x))

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Fricas [A]  time = 0.240511, size = 149, normalized size = 2.13 \[ \frac{{\left (x^{2} - 1\right )} \log \left (\frac{x^{2} + \sqrt{x^{4} - 1} \sqrt{x^{2} - 1} - 1}{x^{2} - 1}\right ) -{\left (x^{2} - 1\right )} \log \left (-\frac{x^{2} - \sqrt{x^{4} - 1} \sqrt{x^{2} - 1} - 1}{x^{2} - 1}\right ) + 2 \, \sqrt{x^{4} - 1}{\left ({\left (x^{2} - 1\right )} \operatorname{arcsec}\left (x\right ) - \sqrt{x^{2} - 1}\right )}}{4 \,{\left (x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215627, size = 70, normalized size = 1. \[ \frac{1}{2} \, \sqrt{x^{4} - 1} \arccos \left (\frac{1}{x}\right ) - \frac{2 \, \sqrt{x^{2} + 1} -{\rm ln}\left (\sqrt{x^{2} + 1} + 1\right ) +{\rm ln}\left (\sqrt{x^{2} + 1} - 1\right )}{4 \,{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^4 - 1)*arccos(1/x) - 1/4*(2*sqrt(x^2 + 1) - ln(sqrt(x^2 + 1) + 1) + l
n(sqrt(x^2 + 1) - 1))/sign(x)