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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.085914, 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, Rules used = {271, 192, 191, 5239, 12, 1259, 453, 206} \[ -\frac{1}{\sqrt{x^2}}-\frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{x^2-1}}-\frac{4 x \csc ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}}+\frac{\csc ^{-1}(x)}{x \left (x^2-1\right )^{3/2}}-\frac{11 x \tanh ^{-1}(x)}{6 \sqrt{x^2}} \]

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*ArcTanh[x])/(6*Sqrt[x^2])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 5239

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx &=\frac{\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac{4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{-1+x^2}}+\frac{x \int \frac{3-12 x^2+8 x^4}{3 x^2 \left (1-x^2\right )^2} \, dx}{\sqrt{x^2}}\\ &=\frac{\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac{4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{-1+x^2}}+\frac{x \int \frac{3-12 x^2+8 x^4}{x^2 \left (1-x^2\right )^2} \, dx}{3 \sqrt{x^2}}\\ &=-\frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac{4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{-1+x^2}}-\frac{x \int \frac{-6+17 x^2}{x^2 \left (1-x^2\right )} \, dx}{6 \sqrt{x^2}}\\ &=-\frac{1}{\sqrt{x^2}}-\frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac{4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{-1+x^2}}-\frac{(11 x) \int \frac{1}{1-x^2} \, dx}{6 \sqrt{x^2}}\\ &=-\frac{1}{\sqrt{x^2}}-\frac{\sqrt{x^2}}{6 \left (1-x^2\right )}+\frac{\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac{4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac{8 x \csc ^{-1}(x)}{3 \sqrt{-1+x^2}}-\frac{11 x \tanh ^{-1}(x)}{6 \sqrt{x^2}}\\ \end{align*}

Mathematica [A]  time = 0.128443, 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))

________________________________________________________________________________________

Maple [C]  time = 0.475, size = 702, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(I*((x^2-1)/x^2)^(1/2)*x-1)/x/(x^2-1)^(1/2)*(3*I*x^2-4*I-4*((x^2-1)/x^2)^(1/2)*x+((x^2-1)/x^2)^(1/2)*x^3)
+1/4/(x^2-1)^(1/2)/x*(x^2-2+2*I*((x^2-1)/x^2)^(1/2)*x)*arccsc(x)+1/2*x*arccsc(x)/(x^2-1)^(1/2)+1/4/(x^2-1)^(1/
2)*(x^2-2-2*I*((x^2-1)/x^2)^(1/2)*x)*arccsc(x)/x+1/4*x^3/(x^2-1)^(1/2)/(I*x^2-2*((x^2-1)/x^2)^(1/2)*x-2*I)-1/2
4*x^5*(((x^2-1)/x^2)^(1/2)*x+I)/(x^2-1)^(1/2)/(I*((x^2-1)/x^2)^(1/2)*x^5-5*I*((x^2-1)/x^2)^(1/2)*x^3-3*x^4+4*I
*((x^2-1)/x^2)^(1/2)*x+7*x^2-4)+1/24*x*(5*I*x^4-20*I*x^2-12*((x^2-1)/x^2)^(1/2)*x^3+((x^2-1)/x^2)^(1/2)*x^5+16
*I+16*((x^2-1)/x^2)^(1/2)*x)/(x^2-1)^(1/2)/(I*((x^2-1)/x^2)^(1/2)*x^5-5*I*((x^2-1)/x^2)^(1/2)*x^3-3*x^4+4*I*((
x^2-1)/x^2)^(1/2)*x+7*x^2-4)+2/3*(x^2-1)^(1/2)*x^3/(x^4-2*x^2+1)*arccsc(x)+1/2*(x^2-1)^(1/2)*x*(x^2-2-2*I*((x^
2-1)/x^2)^(1/2)*x)*arccsc(x)/(x^4-2*x^2+1)+1/2*(x^2-1)^(1/2)*x*(x^2-2+2*I*((x^2-1)/x^2)^(1/2)*x)*arccsc(x)/(x^
4-2*x^2+1)-11/12/(x^2-1)^(1/2)*(((x^2-1)/x^2)^(1/2)*x+I)*ln(I/x+(1-1/x^2)^(1/2)+I)-11/12/(x^2-1)^(1/2)*(((x^2-
1)/x^2)^(1/2)*x-I)*ln(I/x+(1-1/x^2)^(1/2)+I)+11/12/(x^2-1)^(1/2)*(((x^2-1)/x^2)^(1/2)*x+I)*ln(I/x+(1-1/x^2)^(1
/2)-I)+11/12/(x^2-1)^(1/2)*(((x^2-1)/x^2)^(1/2)*x-I)*ln(I/x+(1-1/x^2)^(1/2)-I)

________________________________________________________________________________________

Maxima [B]  time = 2.57892, size = 166, normalized size = 2.37 \begin{align*} \frac{32 \, x^{4} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) -{\left (x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1}{\left (\frac{2 \,{\left (5 \, x^{2} - 6\right )}}{x^{3} - x} + 11 \, \log \left (x + 1\right ) - 11 \, \log \left (x - 1\right )\right )} - 48 \, x^{2} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) + 12 \, \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right )}{12 \,{\left (x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(32*x^4*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) - (x^3 - x)*sqrt(x + 1)*sqrt(x - 1)*(2*(5*x^2 - 6)/(x^3 - x)
+ 11*log(x + 1) - 11*log(x - 1)) - 48*x^2*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) + 12*arctan2(1, sqrt(x + 1)*sqrt
(x - 1)))/((x^3 - x)*sqrt(x + 1)*sqrt(x - 1))

________________________________________________________________________________________

Fricas [A]  time = 2.53592, size = 223, normalized size = 3.19 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x)/x^2/(x^2-1)^(5/2),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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.13022, size = 142, normalized size = 2.03 \begin{align*} \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 )}{\mathrm{sgn}\left (x\right )} - \frac{11 \, \log \left ({\left | x + 1 \right |}\right )}{12 \, \mathrm{sgn}\left (x\right )} + \frac{11 \, \log \left ({\left | x - 1 \right |}\right )}{12 \, \mathrm{sgn}\left (x\right )} - \frac{5 \, x^{2} - 6}{6 \,{\left (x^{3} - x\right )} \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x)/x^2/(x^2-1)^(5/2),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))
/sgn(x) - 11/12*log(abs(x + 1))/sgn(x) + 11/12*log(abs(x - 1))/sgn(x) - 1/6*(5*x^2 - 6)/((x^3 - x)*sgn(x))