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

Optimal. Leaf size=82 \[ \frac{x}{6 \sqrt{x^2} \left (1-x^2\right )}-\frac{2 x \log (x)}{3 \sqrt{x^2}}+\frac{x \log \left (x^2-1\right )}{3 \sqrt{x^2}}-\frac{\sec ^{-1}(x)}{\sqrt{x^2-1}}-\frac{\sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}} \]

[Out]

x/(6*Sqrt[x^2]*(1 - x^2)) - ArcSec[x]/(3*(-1 + x^2)^(3/2)) - ArcSec[x]/Sqrt[-1 +
x^2] - (2*x*Log[x])/(3*Sqrt[x^2]) + (x*Log[-1 + x^2])/(3*Sqrt[x^2])

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Rubi [A]  time = 0.158123, antiderivative size = 84, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{x}{6 \sqrt{x^2} \left (1-x^2\right )}-\frac{2 x \log (x)}{3 \sqrt{x^2}}+\frac{x \log \left (1-x^2\right )}{3 \sqrt{x^2}}-\frac{\sec ^{-1}(x)}{\sqrt{x^2-1}}-\frac{\sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

x/(6*Sqrt[x^2]*(1 - x^2)) - ArcSec[x]/(3*(-1 + x^2)^(3/2)) - ArcSec[x]/Sqrt[-1 +
x^2] - (2*x*Log[x])/(3*Sqrt[x^2]) + (x*Log[1 - x^2])/(3*Sqrt[x^2])

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Rubi in Sympy [A]  time = 8.65708, size = 71, normalized size = 0.87 \[ - \frac{x \log{\left (x^{2} \right )}}{3 \sqrt{x^{2}}} + \frac{x \log{\left (- x^{2} + 1 \right )}}{3 \sqrt{x^{2}}} + \frac{x}{6 \left (- x^{2} + 1\right ) \sqrt{x^{2}}} - \frac{\operatorname{asec}{\left (x \right )}}{\sqrt{x^{2} - 1}} - \frac{\operatorname{asec}{\left (x \right )}}{3 \left (x^{2} - 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*log(x**2)/(3*sqrt(x**2)) + x*log(-x**2 + 1)/(3*sqrt(x**2)) + x/(6*(-x**2 + 1)
*sqrt(x**2)) - asec(x)/sqrt(x**2 - 1) - asec(x)/(3*(x**2 - 1)**(3/2))

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Mathematica [A]  time = 0.228939, size = 72, normalized size = 0.88 \[ \frac{-\frac{\left (x^2-1\right ) \left (4 \left (x^2-1\right ) \log (x)-2 \left (x^2-1\right ) \log \left (1-x^2\right )+1\right )}{\sqrt{1-\frac{1}{x^2}} x}-2 \left (3 x^2-2\right ) \sec ^{-1}(x)}{6 \left (x^2-1\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-2 + 3*x^2)*ArcSec[x] - ((-1 + x^2)*(1 + 4*(-1 + x^2)*Log[x] - 2*(-1 + x^2)
*Log[1 - x^2]))/(Sqrt[1 - x^(-2)]*x))/(6*(-1 + x^2)^(3/2))

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Maple [C]  time = 0.615, size = 197, normalized size = 2.4 \[{-{\frac{4\,i}{3}}x{\rm arcsec} \left (x\right )\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{1}{6\,{x}^{2} \left ( 4\,{x}^{6}-11\,{x}^{4}+10\,{x}^{2}-3 \right ) }\sqrt{{x}^{2}-1} \left ( 2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-3\,{x}^{2}+2 \right ) \left ( 8\,{\rm arcsec} \left (x\right ){x}^{4}+2\,i{x}^{4}+3\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-6\,{\rm arcsec} \left (x\right ){x}^{2}-4\,i{x}^{2}-2\,\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+2\,i \right ) }+{\frac{2\,x}{3}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ( \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) ^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/3*I/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*arcsec(x)+1/6*(x^2-1)^(1/2)/x^2*(2*I*
((x^2-1)/x^2)^(1/2)*x^3-2*I*((x^2-1)/x^2)^(1/2)*x-3*x^2+2)*(8*arcsec(x)*x^4+2*I*
x^4+3*((x^2-1)/x^2)^(1/2)*x^3-6*arcsec(x)*x^2-4*I*x^2-2*((x^2-1)/x^2)^(1/2)*x+2*
I)/(4*x^6-11*x^4+10*x^2-3)+2/3/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*ln((1/x+I*(1-
1/x^2)^(1/2))^2-1)

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Maxima [A]  time = 1.433, size = 86, normalized size = 1.05 \[ -\frac{1}{3} \,{\left (\frac{3 \, x^{2}}{{\left (x^{2} - 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (x^{2} - 1\right )}^{\frac{3}{2}}}\right )} \operatorname{arcsec}\left (x\right ) + \frac{1}{3 \,{\left (x^{2} - 1\right )}} - \frac{1}{2 \,{\left (x + 1\right )}{\left (x - 1\right )}} + \frac{1}{3} \, \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) - \frac{2}{3} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*x^2/(x^2 - 1)^(3/2) - 2/(x^2 - 1)^(3/2))*arcsec(x) + 1/3/(x^2 - 1) - 1/2
/((x + 1)*(x - 1)) + 1/3*log(x + 1) + 1/3*log(x - 1) - 2/3*log(x)

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Fricas [A]  time = 0.225422, size = 93, normalized size = 1.13 \[ -\frac{2 \,{\left (3 \, x^{2} - 2\right )} \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) + x^{2} - 2 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x^{2} - 1\right ) + 4 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x\right ) - 1}{6 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*(3*x^2 - 2)*sqrt(x^2 - 1)*arcsec(x) + x^2 - 2*(x^4 - 2*x^2 + 1)*log(x^2
- 1) + 4*(x^4 - 2*x^2 + 1)*log(x) - 1)/(x^4 - 2*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**2-1)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231543, size = 86, normalized size = 1.05 \[ -\frac{{\left (3 \, x^{2} - 2\right )} \arccos \left (\frac{1}{x}\right )}{3 \,{\left (x^{2} - 1\right )}^{\frac{3}{2}}} - \frac{{\rm ln}\left (x^{2}\right )}{3 \,{\rm sign}\left (x\right )} + \frac{{\rm ln}\left ({\left | x^{2} - 1 \right |}\right )}{3 \,{\rm sign}\left (x\right )} - \frac{2 \, x^{2} - 1}{6 \,{\left (x^{2} - 1\right )}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*x^2 - 2)*arccos(1/x)/(x^2 - 1)^(3/2) - 1/3*ln(x^2)/sign(x) + 1/3*ln(abs(
x^2 - 1))/sign(x) - 1/6*(2*x^2 - 1)/((x^2 - 1)*sign(x))