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

3.7.84.1 Optimal result

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

output

-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
 

3.7.84.2 Mathematica [A] (verified)

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

input

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

output

(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*ArcC 
sc[x]])))/(24*Sqrt[1 - x^(-2)]*x)
 

3.7.84.3 Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {5766, 5200, 243, 49, 2009, 5200, 244, 2009, 5156, 15, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (x^2-1\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx\)

\(\Big \downarrow \) 5766

\(\displaystyle -\frac {\sqrt {x^2} \int \left (1-\frac {1}{x^2}\right )^{5/2} x^4 \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}}{x}\)

\(\Big \downarrow \) 5200

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \int \left (1-\frac {1}{x^2}\right )^{3/2} x^2 \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}+\frac {1}{3} \int \left (1-\frac {1}{x^2}\right )^2 x^3d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )\right )}{x}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \int \left (1-\frac {1}{x^2}\right )^{3/2} x^2 \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}+\frac {1}{6} \int \left (1-\frac {1}{x^2}\right )^2 x^2d\frac {1}{x^2}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )\right )}{x}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \int \left (1-\frac {1}{x^2}\right )^{3/2} x^2 \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}+\frac {1}{6} \int \left (x^2-2 x+1\right )d\frac {1}{x^2}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \int \left (1-\frac {1}{x^2}\right )^{3/2} x^2 \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

\(\Big \downarrow \) 5200

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \left (-3 \int \sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}+\int \left (1-\frac {1}{x^2}\right ) xd\frac {1}{x}-x \left (1-\frac {1}{x^2}\right )^{3/2} \arcsin \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

\(\Big \downarrow \) 244

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \left (-3 \int \sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}+\int \left (x-\frac {1}{x}\right )d\frac {1}{x}-x \left (1-\frac {1}{x^2}\right )^{3/2} \arcsin \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \left (-3 \int \sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}-x \left (1-\frac {1}{x^2}\right )^{3/2} \arcsin \left (\frac {1}{x}\right )-\frac {1}{2 x^2}+\log \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

\(\Big \downarrow \) 5156

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \left (-3 \left (\frac {1}{2} \int \frac {\arcsin \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}-\frac {1}{2} \int \frac {1}{x}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )}{2 x}\right )-x \left (1-\frac {1}{x^2}\right )^{3/2} \arcsin \left (\frac {1}{x}\right )-\frac {1}{2 x^2}+\log \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \left (-3 \left (\frac {1}{2} \int \frac {\arcsin \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )}{2 x}-\frac {1}{4 x^2}\right )-x \left (1-\frac {1}{x^2}\right )^{3/2} \arcsin \left (\frac {1}{x}\right )-\frac {1}{2 x^2}+\log \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {\sqrt {x^2} \left (-\frac {5}{3} \left (-x \left (1-\frac {1}{x^2}\right )^{3/2} \arcsin \left (\frac {1}{x}\right )-3 \left (\frac {\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )}{2 x}+\frac {1}{4} \arcsin \left (\frac {1}{x}\right )^2-\frac {1}{4 x^2}\right )-\frac {1}{2 x^2}+\log \left (\frac {1}{x}\right )\right )-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} x^3 \arcsin \left (\frac {1}{x}\right )+\frac {1}{6} \left (\frac {1}{x^2}-2 \log \left (\frac {1}{x^2}\right )-x\right )\right )}{x}\)

input

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

output

-((Sqrt[x^2]*(-1/3*((1 - x^(-2))^(5/2)*x^3*ArcSin[x^(-1)]) + (x^(-2) - x - 
 2*Log[x^(-2)])/6 - (5*(-1/2*1/x^2 - (1 - x^(-2))^(3/2)*x*ArcSin[x^(-1)] - 
 3*(-1/4*1/x^2 + (Sqrt[1 - x^(-2)]*ArcSin[x^(-1)])/(2*x) + ArcSin[x^(-1)]^ 
2/4) + Log[x^(-1)]))/3))/x)
 

3.7.84.3.1 Defintions of rubi rules used

rule 15

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

rule 49

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}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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 5156

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcSin[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] - Simp[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 5200

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*ArcS 
in[c*x])^n/(f*(m + 1))), x] + (-Simp[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] - Simp[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 5766

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[-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]
 

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

input

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

output

-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*cs 
gn(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)
 

3.7.84.5 Fricas [A] (verification not implemented)

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

input

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

output

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
 

3.7.84.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.7.84.7 Maxima [F]

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

input

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

output

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

3.7.84.8 Giac [F]

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

input

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

output

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

3.7.84.9 Mupad [F(-1)]

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

input

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

output

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