\(\int \frac {(-1+x^2)^{5/2} \csc ^{-1}(x)}{x^3} \, dx\) [684]

Optimal result

Integrand size = 15, antiderivative size = 106 \[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\frac {3+2 x^4}{12 x \sqrt {x^2}}-\frac {5 \sqrt {-1+x^2} \csc ^{-1}(x)}{2 x^2}-\frac {5 \left (-1+x^2\right )^{3/2} \csc ^{-1}(x)}{3 x^2}+\frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{3 x^2}-\frac {5 x \csc ^{-1}(x)^2}{4 \sqrt {x^2}}-\frac {7 x \log (x)}{3 \sqrt {x^2}} \]

[Out]

-5/3*(x^2-1)^(3/2)*arccsc(x)/x^2+1/3*(x^2-1)^(5/2)*arccsc(x)/x^2+1/12*(2*x^4+3)/x/(x^2)^(1/2)-5/4*x*arccsc(x)^
2/(x^2)^(1/2)-7/3*x*ln(x)/(x^2)^(1/2)-5/2*arccsc(x)*(x^2-1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5351, 4785, 4741, 4737, 30, 14, 272, 45} \[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\frac {x \sqrt {x^2}}{6}-\frac {7 \sqrt {x^2} \log (x)}{3 x}+\frac {1}{3} \left (x^2\right )^{3/2} \left (1-\frac {1}{x^2}\right )^{5/2} \csc ^{-1}(x)-\frac {5}{3} \sqrt {x^2} \left (1-\frac {1}{x^2}\right )^{3/2} \csc ^{-1}(x)-\frac {5 \sqrt {1-\frac {1}{x^2}} \csc ^{-1}(x)}{2 \sqrt {x^2}}-\frac {5 \sqrt {x^2} \csc ^{-1}(x)^2}{4 x}+\frac {\sqrt {x^2}}{4 x^3} \]

[In]

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

[Out]

Sqrt[x^2]/(4*x^3) + (x*Sqrt[x^2])/6 - (5*Sqrt[1 - x^(-2)]*ArcCsc[x])/(2*Sqrt[x^2]) - (5*(1 - x^(-2))^(3/2)*Sqr
t[x^2]*ArcCsc[x])/3 + ((1 - x^(-2))^(5/2)*(x^2)^(3/2)*ArcCsc[x])/3 - (5*Sqrt[x^2]*ArcCsc[x]^2)/(4*x) - (7*Sqrt
[x^2]*Log[x])/(3*x)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[x*(a + b*ArcSin[c*x])^(
n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4785

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c
^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c,
d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 5351

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[-Sqrt[x
^2]/x, Subst[Int[(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1))), x], x, 1/x], x] /; FreeQ[{a, b, c, d
, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {\left (1-x^2\right )^{5/2} \arcsin (x)}{x^4} \, dx,x,\frac {1}{x}\right )}{x} \\ & = \frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^3} \, dx,x,\frac {1}{x}\right )}{3 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \arcsin (x)}{x^2} \, dx,x,\frac {1}{x}\right )}{3 x} \\ & = -\frac {5}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \csc ^{-1}(x)+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {(1-x)^2}{x^2} \, dx,x,\frac {1}{x^2}\right )}{6 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,\frac {1}{x}\right )}{3 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \sqrt {1-x^2} \arcsin (x) \, dx,x,\frac {1}{x}\right )}{x} \\ & = -\frac {5 \sqrt {1-\frac {1}{x^2}} \csc ^{-1}(x)}{2 \sqrt {x^2}}-\frac {5}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \csc ^{-1}(x)+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {\sqrt {x^2} \text {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {2}{x}\right ) \, dx,x,\frac {1}{x^2}\right )}{6 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,\frac {1}{x}\right )}{3 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int x \, dx,x,\frac {1}{x}\right )}{2 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{2 x} \\ & = \frac {\sqrt {x^2}}{4 x^3}+\frac {x \sqrt {x^2}}{6}-\frac {5 \sqrt {1-\frac {1}{x^2}} \csc ^{-1}(x)}{2 \sqrt {x^2}}-\frac {5}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \csc ^{-1}(x)+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {5 \sqrt {x^2} \csc ^{-1}(x)^2}{4 x}-\frac {7 \sqrt {x^2} \log (x)}{3 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\frac {\sqrt {-1+x^2} \left (4 x^2-30 \csc ^{-1}(x)^2-3 \cos \left (2 \csc ^{-1}(x)\right )+48 \log \left (\frac {1}{x}\right )-8 \log (x)+\csc ^{-1}(x) \left (8 \sqrt {1-\frac {1}{x^2}} x \left (-7+x^2\right )-6 \sin \left (2 \csc ^{-1}(x)\right )\right )\right )}{24 \sqrt {1-\frac {1}{x^2}} x} \]

[In]

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

[Out]

(Sqrt[-1 + x^2]*(4*x^2 - 30*ArcCsc[x]^2 - 3*Cos[2*ArcCsc[x]] + 48*Log[x^(-1)] - 8*Log[x] + ArcCsc[x]*(8*Sqrt[1
- x^(-2)]*x*(-7 + x^2) - 6*Sin[2*ArcCsc[x]])))/(24*Sqrt[1 - x^(-2)]*x)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.50 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.58

method	result	size
default	\(-\frac {5 \,\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \operatorname {arccsc}\left (x \right )^{2}}{4}+\frac {\left (-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +i x^{2}-2 i\right ) \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (2 \,\operatorname {arccsc}\left (x \right )+i\right )}{16 x^{2}}-\frac {\left (2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +i x^{2}-2 i\right ) \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (-i+2 \,\operatorname {arccsc}\left (x \right )\right )}{16 x^{2}}-\frac {14 i \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \operatorname {arccsc}\left (x \right )}{3}+\frac {\left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-7 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +7 i\right ) \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (2 \,\operatorname {arccsc}\left (x \right ) x^{4}+\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-30 \,\operatorname {arccsc}\left (x \right ) x^{2}-7 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +126 \,\operatorname {arccsc}\left (x \right )-7 i\right )}{6 x^{4}-90 x^{2}+378}+\frac {7 \,\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \ln \left ({\left (\frac {i}{x}+\sqrt {1-\frac {1}{x^{2}}}\right )}^{2}-1\right )}{3}\)	\(274\)

[In]

int((x^2-1)^(5/2)*arccsc(x)/x^3,x,method=_RETURNVERBOSE)

[Out]

-5/4*csgn(x*(1-1/x^2)^(1/2))*arccsc(x)^2+1/16*(-2*((x^2-1)/x^2)^(1/2)*x+I*x^2-2*I)/x^2*csgn(x*(1-1/x^2)^(1/2))
*(2*arccsc(x)+I)-1/16*(2*((x^2-1)/x^2)^(1/2)*x+I*x^2-2*I)*csgn(x*(1-1/x^2)^(1/2))*(-I+2*arccsc(x))/x^2-14/3*I*
csgn(x*(1-1/x^2)^(1/2))*arccsc(x)+1/6*(((x^2-1)/x^2)^(1/2)*x^3-7*((x^2-1)/x^2)^(1/2)*x+7*I)*csgn(x*(1-1/x^2)^(
1/2))*(2*arccsc(x)*x^4+((x^2-1)/x^2)^(1/2)*x^3-30*arccsc(x)*x^2-7*((x^2-1)/x^2)^(1/2)*x+126*arccsc(x)-7*I)/(x^
4-15*x^2+63)+7/3*csgn(x*(1-1/x^2)^(1/2))*ln((I/x+(1-1/x^2)^(1/2))^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.48 \[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\frac {2 \, x^{4} - 15 \, x^{2} \operatorname {arccsc}\left (x\right )^{2} - 28 \, x^{2} \log \left (x\right ) + 2 \, {\left (2 \, x^{4} - 14 \, x^{2} - 3\right )} \sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right ) + 3}{12 \, x^{2}} \]

[In]

integrate((x^2-1)^(5/2)*arccsc(x)/x^3,x, algorithm="fricas")

[Out]

1/12*(2*x^4 - 15*x^2*arccsc(x)^2 - 28*x^2*log(x) + 2*(2*x^4 - 14*x^2 - 3)*sqrt(x^2 - 1)*arccsc(x) + 3)/x^2

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\text {Timed out} \]

[In]

integrate((x**2-1)**(5/2)*acsc(x)/x**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {5}{2}} \operatorname {arccsc}\left (x\right )}{x^{3}} \,d x } \]

[In]

integrate((x^2-1)^(5/2)*arccsc(x)/x^3,x, algorithm="maxima")

[Out]

integrate((x^2 - 1)^(5/2)*arccsc(x)/x^3, x)

Giac [F]

\[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {5}{2}} \operatorname {arccsc}\left (x\right )}{x^{3}} \,d x } \]

[In]

integrate((x^2-1)^(5/2)*arccsc(x)/x^3,x, algorithm="giac")

[Out]

integrate((x^2 - 1)^(5/2)*arccsc(x)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{x}\right )\,{\left (x^2-1\right )}^{5/2}}{x^3} \,d x \]

[In]

int((asin(1/x)*(x^2 - 1)^(5/2))/x^3,x)

[Out]

int((asin(1/x)*(x^2 - 1)^(5/2))/x^3, x)