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

Optimal. Leaf size=107 \[ \frac{i \sqrt{x^2} \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )}{x}-\frac{i \sqrt{x^2} \text{PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )}{x}-\frac{1}{\sqrt{x^2}}-\frac{\sqrt{x^2-1} \sec ^{-1}(x)}{x}-\frac{2 i \sqrt{x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x} \]

[Out]

-(1/Sqrt[x^2]) - (Sqrt[-1 + x^2]*ArcSec[x])/x - ((2*I)*Sqrt[x^2]*ArcSec[x]*ArcTa
n[E^(I*ArcSec[x])])/x + (I*Sqrt[x^2]*PolyLog[2, (-I)*E^(I*ArcSec[x])])/x - (I*Sq
rt[x^2]*PolyLog[2, I*E^(I*ArcSec[x])])/x

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Rubi [A]  time = 0.291478, antiderivative size = 116, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{i \sqrt{x^2} \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )}{x}-\frac{i \sqrt{x^2} \text{PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )}{x}-\frac{1}{\sqrt{x^2}}-\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \sec ^{-1}(x)}{x}-\frac{2 i \sqrt{x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

-(1/Sqrt[x^2]) - (Sqrt[1 - x^(-2)]*Sqrt[x^2]*ArcSec[x])/x - ((2*I)*Sqrt[x^2]*Arc
Sec[x]*ArcTan[E^(I*ArcSec[x])])/x + (I*Sqrt[x^2]*PolyLog[2, (-I)*E^(I*ArcSec[x])
])/x - (I*Sqrt[x^2]*PolyLog[2, I*E^(I*ArcSec[x])])/x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{x} + \frac{\sqrt{x^{2}} \int ^{\operatorname{acos}{\left (\frac{1}{x} \right )}} \frac{x}{\cos{\left (x \right )}}\, dx}{x} - \frac{\sqrt{x^{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(1 - 1/x**2)*sqrt(x**2)*acos(1/x)/x + sqrt(x**2)*Integral(x/cos(x), (x, aco
s(1/x)))/x - sqrt(x**2)/x**2

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Mathematica [A]  time = 0.212723, size = 116, normalized size = 1.08 \[ -\frac{\sqrt{1-\frac{1}{x^2}} \left (-i x \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )+i x \text{PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )+\sqrt{1-\frac{1}{x^2}} x \sec ^{-1}(x)-x \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right )+x \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right )+1\right )}{\sqrt{x^2-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((Sqrt[1 - x^(-2)]*(1 + Sqrt[1 - x^(-2)]*x*ArcSec[x] - x*ArcSec[x]*Log[1 - I*E^
(I*ArcSec[x])] + x*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])] - I*x*PolyLog[2, (-I)*E^
(I*ArcSec[x])] + I*x*PolyLog[2, I*E^(I*ArcSec[x])]))/Sqrt[-1 + x^2])

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Maple [C]  time = 0.263, size = 199, normalized size = 1.9 \[ -{\frac{{\rm arcsec} \left (x\right )+i}{2\,x} \left ( -i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{{\rm arcsec} \left (x\right )-i}{2\,x} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{x\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}} \left ({\rm arcsec} \left (x\right )\ln \left ( 1+i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) -{\rm arcsec} \left (x\right )\ln \left ( 1-i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) -i{\it dilog} \left ( 1+i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) +i{\it dilog} \left ( 1-i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(x^2-1)^(1/2)/x*(-I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(arcsec(x)+I)-1/2/(x^2-1)^
(1/2)*(I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(arcsec(x)-I)/x-((x^2-1)/x^2)^(1/2)*x*(arc
sec(x)*ln(1+I*(1/x+I*(1-1/x^2)^(1/2)))-arcsec(x)*ln(1-I*(1/x+I*(1-1/x^2)^(1/2)))
-I*dilog(1+I*(1/x+I*(1-1/x^2)^(1/2)))+I*dilog(1-I*(1/x+I*(1-1/x^2)^(1/2))))/(x^2
-1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(x^2 - 1)*arcsec(x)/x^2, x)

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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)**(1/2)/x**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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