3.689 \(\int \frac{x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=175 \[ \frac{5 i \sqrt{x^2} \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac{5 i \sqrt{x^2} \text{PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )}{2 x}+\frac{\sqrt{x^2} \left (2-3 x^2\right )}{6 \left (x^2-1\right )}-\frac{5 x \sec ^{-1}(x)}{2 \sqrt{x^2-1}}-\frac{13}{6} \coth ^{-1}\left (\sqrt{x^2}\right )-\frac{5 i \sqrt{x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x}+\frac{x^5 \sec ^{-1}(x)}{2 \left (x^2-1\right )^{3/2}}-\frac{5 x^3 \sec ^{-1}(x)}{6 \left (x^2-1\right )^{3/2}} \]

[Out]

(Sqrt[x^2]*(2 - 3*x^2))/(6*(-1 + x^2)) - (13*ArcCoth[Sqrt[x^2]])/6 - (5*x^3*ArcS
ec[x])/(6*(-1 + x^2)^(3/2)) + (x^5*ArcSec[x])/(2*(-1 + x^2)^(3/2)) - (5*x*ArcSec
[x])/(2*Sqrt[-1 + x^2]) - ((5*I)*Sqrt[x^2]*ArcSec[x]*ArcTan[E^(I*ArcSec[x])])/x
+ (((5*I)/2)*Sqrt[x^2]*PolyLog[2, (-I)*E^(I*ArcSec[x])])/x - (((5*I)/2)*Sqrt[x^2
]*PolyLog[2, I*E^(I*ArcSec[x])])/x

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Rubi [A]  time = 0.46795, antiderivative size = 232, normalized size of antiderivative = 1.33, number of steps used = 16, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733 \[ \frac{5 i \sqrt{x^2} \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac{5 i \sqrt{x^2} \text{PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )}{2 x}+\frac{\sqrt{x^2}}{4 \left (1-\frac{1}{x^2}\right )}-\frac{3 \sqrt{x^2}}{4}-\frac{5}{12 \left (1-\frac{1}{x^2}\right ) \sqrt{x^2}}+\frac{x \sqrt{x^2} \sec ^{-1}(x)}{2 \left (1-\frac{1}{x^2}\right )^{3/2}}-\frac{5 \sqrt{x^2} \sec ^{-1}(x)}{2 \sqrt{1-\frac{1}{x^2}} x}-\frac{5 \sqrt{x^2} \sec ^{-1}(x)}{6 \left (1-\frac{1}{x^2}\right )^{3/2} x}-\frac{13 \sqrt{x^2} \coth ^{-1}(x)}{6 x}-\frac{5 i \sqrt{x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x} \]

Warning: Unable to verify antiderivative.

[In]  Int[(x^6*ArcSec[x])/(-1 + x^2)^(5/2),x]

[Out]

-5/(12*(1 - x^(-2))*Sqrt[x^2]) - (3*Sqrt[x^2])/4 + Sqrt[x^2]/(4*(1 - x^(-2))) -
(13*Sqrt[x^2]*ArcCoth[x])/(6*x) - (5*Sqrt[x^2]*ArcSec[x])/(6*(1 - x^(-2))^(3/2)*
x) - (5*Sqrt[x^2]*ArcSec[x])/(2*Sqrt[1 - x^(-2)]*x) + (x*Sqrt[x^2]*ArcSec[x])/(2
*(1 - x^(-2))^(3/2)) - ((5*I)*Sqrt[x^2]*ArcSec[x]*ArcTan[E^(I*ArcSec[x])])/x + (
((5*I)/2)*Sqrt[x^2]*PolyLog[2, (-I)*E^(I*ArcSec[x])])/x - (((5*I)/2)*Sqrt[x^2]*P
olyLog[2, I*E^(I*ArcSec[x])])/x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{2 \left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}}} - \frac{3 \sqrt{x^{2}}}{4} + \frac{\sqrt{x^{2}}}{4 \left (1 - \frac{1}{x^{2}}\right )} - \frac{13 \sqrt{x^{2}} \operatorname{atanh}{\left (\frac{1}{x} \right )}}{6 x} + \frac{5 \sqrt{x^{2}} \int ^{\operatorname{acos}{\left (\frac{1}{x} \right )}} \frac{x}{\cos{\left (x \right )}}\, dx}{2 x} - \frac{5 \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{2 x \sqrt{1 - \frac{1}{x^{2}}}} - \frac{5 \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{6 x \left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}}} - \frac{5 \sqrt{x^{2}}}{12 x^{2} \left (1 - \frac{1}{x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*asec(x)/(x**2-1)**(5/2),x)

[Out]

x*sqrt(x**2)*acos(1/x)/(2*(1 - 1/x**2)**(3/2)) - 3*sqrt(x**2)/4 + sqrt(x**2)/(4*
(1 - 1/x**2)) - 13*sqrt(x**2)*atanh(1/x)/(6*x) + 5*sqrt(x**2)*Integral(x/cos(x),
(x, acos(1/x)))/(2*x) - 5*sqrt(x**2)*acos(1/x)/(2*x*sqrt(1 - 1/x**2)) - 5*sqrt(
x**2)*acos(1/x)/(6*x*(1 - 1/x**2)**(3/2)) - 5*sqrt(x**2)/(12*x**2*(1 - 1/x**2))

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Mathematica [B]  time = 2.77508, size = 383, normalized size = 2.19 \[ -\frac{x^5 \left (-60 i \sqrt{1-\frac{1}{x^2}} \sin ^2\left (2 \sec ^{-1}(x)\right ) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )+60 i \sqrt{1-\frac{1}{x^2}} \sin ^2\left (2 \sec ^{-1}(x)\right ) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )-30 \sqrt{1-\frac{1}{x^2}} \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right )+30 \sqrt{1-\frac{1}{x^2}} \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right )-26 \sqrt{1-\frac{1}{x^2}} \log \left (\sin \left (\frac{1}{2} \sec ^{-1}(x)\right )\right )+26 \sqrt{1-\frac{1}{x^2}} \log \left (\cos \left (\frac{1}{2} \sec ^{-1}(x)\right )\right )+22 \sec ^{-1}(x)+40 \sec ^{-1}(x) \cos \left (2 \sec ^{-1}(x)\right )-30 \sec ^{-1}(x) \cos \left (4 \sec ^{-1}(x)\right )+16 \sin \left (2 \sec ^{-1}(x)\right )-4 \sin \left (4 \sec ^{-1}(x)\right )-15 \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right ) \sin \left (3 \sec ^{-1}(x)\right )+15 \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right ) \sin \left (3 \sec ^{-1}(x)\right )-13 \sin \left (3 \sec ^{-1}(x)\right ) \log \left (\sin \left (\frac{1}{2} \sec ^{-1}(x)\right )\right )+15 \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right ) \sin \left (5 \sec ^{-1}(x)\right )-15 \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right ) \sin \left (5 \sec ^{-1}(x)\right )+13 \sin \left (5 \sec ^{-1}(x)\right ) \log \left (\sin \left (\frac{1}{2} \sec ^{-1}(x)\right )\right )+13 \sin \left (3 \sec ^{-1}(x)\right ) \log \left (\cos \left (\frac{1}{2} \sec ^{-1}(x)\right )\right )-13 \sin \left (5 \sec ^{-1}(x)\right ) \log \left (\cos \left (\frac{1}{2} \sec ^{-1}(x)\right )\right )\right )}{96 \left (x^2-1\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^6*ArcSec[x])/(-1 + x^2)^(5/2),x]

[Out]

-(x^5*(22*ArcSec[x] + 40*ArcSec[x]*Cos[2*ArcSec[x]] - 30*ArcSec[x]*Cos[4*ArcSec[
x]] - 30*Sqrt[1 - x^(-2)]*ArcSec[x]*Log[1 - I*E^(I*ArcSec[x])] + 30*Sqrt[1 - x^(
-2)]*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])] + 26*Sqrt[1 - x^(-2)]*Log[Cos[ArcSec[x
]/2]] - 26*Sqrt[1 - x^(-2)]*Log[Sin[ArcSec[x]/2]] + 16*Sin[2*ArcSec[x]] - (60*I)
*Sqrt[1 - x^(-2)]*PolyLog[2, (-I)*E^(I*ArcSec[x])]*Sin[2*ArcSec[x]]^2 + (60*I)*S
qrt[1 - x^(-2)]*PolyLog[2, I*E^(I*ArcSec[x])]*Sin[2*ArcSec[x]]^2 - 15*ArcSec[x]*
Log[1 - I*E^(I*ArcSec[x])]*Sin[3*ArcSec[x]] + 15*ArcSec[x]*Log[1 + I*E^(I*ArcSec
[x])]*Sin[3*ArcSec[x]] + 13*Log[Cos[ArcSec[x]/2]]*Sin[3*ArcSec[x]] - 13*Log[Sin[
ArcSec[x]/2]]*Sin[3*ArcSec[x]] - 4*Sin[4*ArcSec[x]] + 15*ArcSec[x]*Log[1 - I*E^(
I*ArcSec[x])]*Sin[5*ArcSec[x]] - 15*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])]*Sin[5*A
rcSec[x]] - 13*Log[Cos[ArcSec[x]/2]]*Sin[5*ArcSec[x]] + 13*Log[Sin[ArcSec[x]/2]]
*Sin[5*ArcSec[x]]))/(96*(-1 + x^2)^(3/2))

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Maple [C]  time = 0.675, size = 240, normalized size = 1.4 \[{\frac{x}{6\,{x}^{4}-12\,{x}^{2}+6}\sqrt{{x}^{2}-1} \left ( 3\,{\rm arcsec} \left (x\right ){x}^{4}-3\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-20\,{\rm arcsec} \left (x\right ){x}^{2}+2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+15\,{\rm arcsec} \left (x\right ) \right ) }+{{\frac{i}{6}}x\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}} \left ( 15\,i{\rm arcsec} \left (x\right )\ln \left ( 1+i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) -15\,i{\rm arcsec} \left (x\right )\ln \left ( 1-i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) -13\,i\ln \left ({x}^{-1}+i\sqrt{1-{x}^{-2}}-1 \right ) +13\,i\ln \left ( 1+{x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) +15\,{\it dilog} \left ( 1+i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) -15\,{\it dilog} \left ( 1-i \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) \right ) \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*arcsec(x)/(x^2-1)^(5/2),x)

[Out]

1/6*(x^2-1)^(1/2)*x/(x^4-2*x^2+1)*(3*arcsec(x)*x^4-3*((x^2-1)/x^2)^(1/2)*x^3-20*
arcsec(x)*x^2+2*((x^2-1)/x^2)^(1/2)*x+15*arcsec(x))+1/6*I/(x^2-1)^(1/2)*((x^2-1)
/x^2)^(1/2)*x*(15*I*arcsec(x)*ln(1+I*(1/x+I*(1-1/x^2)^(1/2)))-15*I*arcsec(x)*ln(
1-I*(1/x+I*(1-1/x^2)^(1/2)))-13*I*ln(1/x+I*(1-1/x^2)^(1/2)-1)+13*I*ln(1+1/x+I*(1
-1/x^2)^(1/2))+15*dilog(1+I*(1/x+I*(1-1/x^2)^(1/2)))-15*dilog(1-I*(1/x+I*(1-1/x^
2)^(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(x^6*arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6} \operatorname{arcsec}\left (x\right )}{{\left (x^{4} - 2 \, x^{2} + 1\right )} \sqrt{x^{2} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6*arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="fricas")

[Out]

integral(x^6*arcsec(x)/((x^4 - 2*x^2 + 1)*sqrt(x^2 - 1)), 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(x**6*asec(x)/(x**2-1)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6} \operatorname{arcsec}\left (x\right )}{{\left (x^{2} - 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6*arcsec(x)/(x^2 - 1)^(5/2),x, algorithm="giac")

[Out]

integrate(x^6*arcsec(x)/(x^2 - 1)^(5/2), x)