3.691 \(\int \frac{\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{1}{\sqrt{x^2}}+\frac{\sqrt{x^2}}{6 \left (x^2-1\right )}-\frac{11}{6} \coth ^{-1}\left (\sqrt{x^2}\right )+\frac{\left (8 x^4-12 x^2+3\right ) \csc ^{-1}(x)}{3 x \left (x^2-1\right )^{3/2}} \]

[Out]

-(1/Sqrt[x^2]) + Sqrt[x^2]/(6*(-1 + x^2)) - (11*ArcCoth[Sqrt[x^2]])/6 + ((3 - 12
*x^2 + 8*x^4)*ArcCsc[x])/(3*x*(-1 + x^2)^(3/2))

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Rubi [A]  time = 0.165217, antiderivative size = 91, normalized size of antiderivative = 1.3, number of steps used = 5, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ -\frac{1}{\sqrt{x^2}}-\frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{x^2-1}}-\frac{4 x \csc ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}}+\frac{\csc ^{-1}(x)}{x \left (x^2-1\right )^{3/2}}-\frac{11 x \tanh ^{-1}(x)}{6 \sqrt{x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(1/Sqrt[x^2]) - Sqrt[x^2]/(6*(1 - x^2)) + ArcCsc[x]/(x*(-1 + x^2)^(3/2)) - (4*x
*ArcCsc[x])/(3*(-1 + x^2)^(3/2)) + (8*x*ArcCsc[x])/(3*Sqrt[-1 + x^2]) - (11*x*Ar
cTanh[x])/(6*Sqrt[x^2])

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Rubi in Sympy [A]  time = 10.8913, size = 87, normalized size = 1.24 \[ - \frac{x^{2}}{6 \left (- x^{2} + 1\right ) \sqrt{x^{2}}} - \frac{11 x \operatorname{atanh}{\left (x \right )}}{6 \sqrt{x^{2}}} + \frac{8 x \operatorname{acsc}{\left (x \right )}}{3 \sqrt{x^{2} - 1}} - \frac{4 x \operatorname{acsc}{\left (x \right )}}{3 \left (x^{2} - 1\right )^{\frac{3}{2}}} - \frac{1}{\sqrt{x^{2}}} + \frac{\operatorname{acsc}{\left (x \right )}}{x \left (x^{2} - 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2/(6*(-x**2 + 1)*sqrt(x**2)) - 11*x*atanh(x)/(6*sqrt(x**2)) + 8*x*acsc(x)/(3
*sqrt(x**2 - 1)) - 4*x*acsc(x)/(3*(x**2 - 1)**(3/2)) - 1/sqrt(x**2) + acsc(x)/(x
*(x**2 - 1)**(3/2))

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Mathematica [A]  time = 0.177279, size = 79, normalized size = 1.13 \[ \frac{\sqrt{1-\frac{1}{x^2}} x \left (-10 x^2+11 \left (x^2-1\right ) x \log (1-x)-11 \left (x^2-1\right ) x \log (x+1)+12\right )+4 \left (8 x^4-12 x^2+3\right ) \csc ^{-1}(x)}{12 x \left (x^2-1\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(3 - 12*x^2 + 8*x^4)*ArcCsc[x] + Sqrt[1 - x^(-2)]*x*(12 - 10*x^2 + 11*x*(-1 +
x^2)*Log[1 - x] - 11*x*(-1 + x^2)*Log[1 + x]))/(12*x*(-1 + x^2)^(3/2))

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Maple [C]  time = 0.48, size = 203, normalized size = 2.9 \[{\frac{{\rm arccsc} \left (x\right )+i}{2\,x} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{{\rm arccsc} \left (x\right )-i}{2\,x} \left ( -i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{x}{6\,{x}^{4}-12\,{x}^{2}+6}\sqrt{{x}^{2}-1} \left ( 10\,{\rm arccsc} \left (x\right ){x}^{2}+\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-12\,{\rm arccsc} \left (x\right ) \right ) }+{\frac{11\,x}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ({\frac{i}{x}}+\sqrt{1-{x}^{-2}}-i \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{11\,x}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ({\frac{i}{x}}+\sqrt{1-{x}^{-2}}+i \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccsc(x)/x^2/(x^2-1)^(5/2),x)

[Out]

1/2/(x^2-1)^(1/2)/x*(I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(arccsc(x)+I)+1/2/(x^2-1)^(1
/2)*(-I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(arccsc(x)-I)/x+1/6*(x^2-1)^(1/2)*x/(x^4-2*
x^2+1)*(10*arccsc(x)*x^2+((x^2-1)/x^2)^(1/2)*x-12*arccsc(x))+11/6/(x^2-1)^(1/2)*
((x^2-1)/x^2)^(1/2)*x*ln(I/x+(1-1/x^2)^(1/2)-I)-11/6/(x^2-1)^(1/2)*((x^2-1)/x^2)
^(1/2)*x*ln(I/x+(1-1/x^2)^(1/2)+I)

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Maxima [A]  time = 1.62524, size = 105, normalized size = 1.5 \[ \frac{1}{3} \,{\left (\frac{8 \, x}{\sqrt{x^{2} - 1}} - \frac{4 \, x}{{\left (x^{2} - 1\right )}^{\frac{3}{2}}} + \frac{3}{{\left (x^{2} - 1\right )}^{\frac{3}{2}} x}\right )} \operatorname{arccsc}\left (x\right ) - \frac{3 \, x^{2} - 2}{2 \,{\left (x^{3} - x\right )}} + \frac{2 \, x}{3 \,{\left (x^{2} - 1\right )}} - \frac{11}{12} \, \log \left (x + 1\right ) + \frac{11}{12} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(8*x/sqrt(x^2 - 1) - 4*x/(x^2 - 1)^(3/2) + 3/((x^2 - 1)^(3/2)*x))*arccsc(x)
- 1/2*(3*x^2 - 2)/(x^3 - x) + 2/3*x/(x^2 - 1) - 11/12*log(x + 1) + 11/12*log(x -
1)

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Fricas [A]  time = 0.255627, size = 109, normalized size = 1.56 \[ -\frac{10 \, x^{4} - 4 \,{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt{x^{2} - 1} \operatorname{arccsc}\left (x\right ) - 22 \, x^{2} + 11 \,{\left (x^{5} - 2 \, x^{3} + x\right )} \log \left (x + 1\right ) - 11 \,{\left (x^{5} - 2 \, x^{3} + x\right )} \log \left (x - 1\right ) + 12}{12 \,{\left (x^{5} - 2 \, x^{3} + x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(10*x^4 - 4*(8*x^4 - 12*x^2 + 3)*sqrt(x^2 - 1)*arccsc(x) - 22*x^2 + 11*(x^
5 - 2*x^3 + x)*log(x + 1) - 11*(x^5 - 2*x^3 + x)*log(x - 1) + 12)/(x^5 - 2*x^3 +
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(acsc(x)/x**2/(x**2-1)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261763, size = 142, normalized size = 2.03 \[ \frac{1}{3} \,{\left (\frac{{\left (5 \, x^{2} - 6\right )} x}{{\left (x^{2} - 1\right )}^{\frac{3}{2}}} + \frac{6}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1}\right )} \arcsin \left (\frac{1}{x}\right ) + \frac{2 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right )}{{\rm sign}\left (x\right )} - \frac{11 \,{\rm ln}\left ({\left | x + 1 \right |}\right )}{12 \,{\rm sign}\left (x\right )} + \frac{11 \,{\rm ln}\left ({\left | x - 1 \right |}\right )}{12 \,{\rm sign}\left (x\right )} - \frac{5 \, x^{2} - 6}{6 \,{\left (x^{3} - x\right )}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((5*x^2 - 6)*x/(x^2 - 1)^(3/2) + 6/((x - sqrt(x^2 - 1))^2 + 1))*arcsin(1/x)
+ 2*arctan(-x + sqrt(x^2 - 1))/sign(x) - 11/12*ln(abs(x + 1))/sign(x) + 11/12*ln
(abs(x - 1))/sign(x) - 1/6*(5*x^2 - 6)/((x^3 - x)*sign(x))