\(\int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 123 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {\left (1-2 a^2\right ) b^2 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}} \] Output:

-1/2*b*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/a/(-a^2+1)/x+1/2*b^2*arccsc(b*x+a)/a^ 
2-1/2*arccsc(b*x+a)/x^2+(-2*a^2+1)*b^2*arctan((a-tan(1/2*arccsc(b*x+a)))/( 
-a^2+1)^(1/2))/a^2/(-a^2+1)^(3/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\frac {\frac {b x (a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}}{a \left (-1+a^2\right )}-\csc ^{-1}(a+b x)+\frac {b^2 x^2 \arcsin \left (\frac {1}{a+b x}\right )}{a^2}+\frac {i \left (-1+2 a^2\right ) b^2 x^2 \log \left (\frac {4 (-1+a) a^2 (1+a) \left (\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}+(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{\left (-1+2 a^2\right ) b^2 x}\right )}{a^2 \left (1-a^2\right )^{3/2}}}{2 x^2} \] Input:

Integrate[ArcCsc[a + b*x]/x^3,x]
 

Output:

((b*x*(a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])/(a*(-1 + 
 a^2)) - ArcCsc[a + b*x] + (b^2*x^2*ArcSin[(a + b*x)^(-1)])/a^2 + (I*(-1 + 
 2*a^2)*b^2*x^2*Log[(4*(-1 + a)*a^2*(1 + a)*((I*(-1 + a^2 + a*b*x))/Sqrt[1 
 - a^2] + (a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]))/((- 
1 + 2*a^2)*b^2*x)])/(a^2*(1 - a^2)^(3/2)))/(2*x^2)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.25, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {5782, 25, 4927, 3042, 4272, 3042, 4407, 3042, 4318, 3042, 3139, 1083, 217}

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.

\(\displaystyle \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx\)

\(\Big \downarrow \) 5782

\(\displaystyle -b^2 \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3 x^3}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 25

\(\displaystyle b^2 \int -\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3 x^3}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 4927

\(\displaystyle -b^2 \left (\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}-\frac {1}{2} \int \frac {1}{b^2 x^2}d\csc ^{-1}(a+b x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^2 \left (\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}-\frac {1}{2} \int \frac {1}{\left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^2}d\csc ^{-1}(a+b x)\right )\)

\(\Big \downarrow \) 4272

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\int -\frac {-a^2-(a+b x) a+1}{b x}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\int \frac {-a^2-\csc \left (\csc ^{-1}(a+b x)\right ) a+1}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 4407

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-2 a^2\right ) \int -\frac {a+b x}{b x}d\csc ^{-1}(a+b x)}{a}+\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-2 a^2\right ) \int \frac {\csc \left (\csc ^{-1}(a+b x)\right )}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}+\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 4318

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}-\frac {\left (1-2 a^2\right ) \int \frac {1}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}-\frac {\left (1-2 a^2\right ) \int \frac {1}{1-a \sin \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 3139

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}-\frac {2 \left (1-2 a^2\right ) \int \frac {1}{\tan ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+1}d\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {4 \left (1-2 a^2\right ) \int \frac {1}{-\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )^2-4 \left (1-a^2\right )}d\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )}{a}+\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-a^2\right ) \csc ^{-1}(a+b x)}{a}-\frac {2 \left (1-2 a^2\right ) \arctan \left (\frac {2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a}{2 \sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}}{a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{2 b^2 x^2}\right )\)

Input:

Int[ArcCsc[a + b*x]/x^3,x]
 

Output:

-(b^2*(ArcCsc[a + b*x]/(2*b^2*x^2) + (((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) 
/(a*(1 - a^2)*b*x) - (((1 - a^2)*ArcCsc[a + b*x])/a - (2*(1 - 2*a^2)*ArcTa 
n[(-2*a + 2*Tan[ArcCsc[a + b*x]/2])/(2*Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])) 
/(a*(1 - a^2)))/2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

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

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a   Int[Csc[e + f* 
x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0]
 

rule 4927

Int[Cot[(c_.) + (d_.)*(x_)]*Csc[(c_.) + (d_.)*(x_)]*(Csc[(c_.) + (d_.)*(x_) 
]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(e + 
f*x)^m)*((a + b*Csc[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b*d*( 
n + 1)))   Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 5782

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csc[x]*Cot 
[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(109)=218\).

Time = 0.50 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.59

method	result	size
parts	\(-\frac {\operatorname {arccsc}\left (b x +a \right )}{2 x^{2}}+\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (\left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{2} b x -2 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{4} b x -b \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) x \left (a^{2}-1\right )^{\frac {3}{2}}+\left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a +3 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{2} b x -b \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) x \right )}{2 \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}} x}\)	\(319\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {arccsc}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{3}-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )-2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{5}+2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{4} \left (b x +a \right )-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a +\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} \left (b x +a \right )-\left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {\left (b x +a \right )^{2}-1}\, a +3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{2} \left (b x +a \right )-a \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )+\ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}} b x}\right )\)	\(457\)
default	\(b^{2} \left (-\frac {\operatorname {arccsc}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{3}-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )-2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{5}+2 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{4} \left (b x +a \right )-\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} a +\arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right ) \left (a^{2}-1\right )^{\frac {3}{2}} \left (b x +a \right )-\left (a^{2}-1\right )^{\frac {3}{2}} \sqrt {\left (b x +a \right )^{2}-1}\, a +3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) a^{2} \left (b x +a \right )-a \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right )+\ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 \left (b x +a \right ) a -2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}} b x}\right )\)	\(457\)

Input:

int(arccsc(b*x+a)/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*arccsc(b*x+a)/x^2+1/2*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*((a^2-1)^(3/2)* 
arctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*a^2*b*x-2*ln(2*(a*b*x+(a^2-1)^(1/2 
)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^4*b*x-b*arctan(1/(b^2*x^2+2*a* 
b*x+a^2-1)^(1/2))*x*(a^2-1)^(3/2)+(a^2-1)^(3/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1 
/2)*a+3*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)* 
a^2*b*x-b*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x 
)*x)/((b^2*x^2+2*a*b*x+a^2-1)/(b*x+a)^2)^(1/2)/(b*x+a)/a^2/(a^2-1)^(5/2)/x
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.48 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\left [\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} - 1} b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{3} - a\right )} b x - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname {arccsc}\left (b x + a\right )}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}, \frac {2 \, {\left (2 \, a^{2} - 1\right )} \sqrt {-a^{2} + 1} b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{3} - a\right )} b x - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname {arccsc}\left (b x + a\right )}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}\right ] \] Input:

integrate(arccsc(b*x+a)/x^3,x, algorithm="fricas")
 

Output:

[1/2*((2*a^2 - 1)*sqrt(a^2 - 1)*b^2*x^2*log((a^2*b*x + a^3 + sqrt(b^2*x^2 
+ 2*a*b*x + a^2 - 1)*(a^2 - sqrt(a^2 - 1)*a - 1) - (a*b*x + a^2 - 1)*sqrt( 
a^2 - 1) - a)/x) - 2*(a^4 - 2*a^2 + 1)*b^2*x^2*arctan(-b*x - a + sqrt(b^2* 
x^2 + 2*a*b*x + a^2 - 1)) + (a^3 - a)*b^2*x^2 + sqrt(b^2*x^2 + 2*a*b*x + a 
^2 - 1)*(a^3 - a)*b*x - (a^6 - 2*a^4 + a^2)*arccsc(b*x + a))/((a^6 - 2*a^4 
 + a^2)*x^2), 1/2*(2*(2*a^2 - 1)*sqrt(-a^2 + 1)*b^2*x^2*arctan(-(sqrt(-a^2 
 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*sqrt(-a^2 + 1))/(a^2 - 1)) - 
 2*(a^4 - 2*a^2 + 1)*b^2*x^2*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^ 
2 - 1)) + (a^3 - a)*b^2*x^2 + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^3 - a)* 
b*x - (a^6 - 2*a^4 + a^2)*arccsc(b*x + a))/((a^6 - 2*a^4 + a^2)*x^2)]
 

Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{3}}\, dx \] Input:

integrate(acsc(b*x+a)/x**3,x)
 

Output:

Integral(acsc(a + b*x)/x**3, x)
 

Maxima [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x^{3}} \,d x } \] Input:

integrate(arccsc(b*x+a)/x^3,x, algorithm="maxima")
 

Output:

-1/2*(2*x^2*integrate(1/2*(b^2*x + a*b)*e^(1/2*log(b*x + a + 1) + 1/2*log( 
b*x + a - 1))/(b^2*x^4 + 2*a*b*x^3 + (a^2 - 1)*x^2 + (b^2*x^4 + 2*a*b*x^3 
+ (a^2 - 1)*x^2)*e^(log(b*x + a + 1) + log(b*x + a - 1))), x) + arctan2(1, 
 sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))/x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (108) = 216\).

Time = 0.17 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.76 \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\frac {1}{2} \, b {\left (\frac {2 \, {\left (2 \, a^{2} b - b\right )} \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - a^{2}\right )} \sqrt {-a^{2} + 1}} + \frac {2 \, {\left ({\left (b x + a\right )} a b {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + b\right )}}{{\left ({\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 2 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1\right )} {\left (a^{3} - a\right )}} - \frac {{\left (\frac {2 \, a b}{b x + a} - b\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a^{2} {\left (\frac {a}{b x + a} - 1\right )}^{2}}\right )} \] Input:

integrate(arccsc(b*x+a)/x^3,x, algorithm="giac")
 

Output:

1/2*b*(2*(2*a^2*b - b)*arctan(((b*x + a)*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 
a)/sqrt(-a^2 + 1))/((a^4 - a^2)*sqrt(-a^2 + 1)) + 2*((b*x + a)*a*b*(sqrt(- 
1/(b*x + a)^2 + 1) - 1) + b)/(((b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^ 
2 + 2*(b*x + a)*a*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 1)*(a^3 - a)) - (2*a*b/ 
(b*x + a) - b)*arcsin(-1/((b*x + a)*(a/(b*x + a) - 1) - a))/(a^2*(a/(b*x + 
 a) - 1)^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \] Input:

int(asin(1/(a + b*x))/x^3,x)
 

Output:

int(asin(1/(a + b*x))/x^3, x)
 

Reduce [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx=\int \frac {\mathit {acsc} \left (b x +a \right )}{x^{3}}d x \] Input:

int(acsc(b*x+a)/x^3,x)
 

Output:

int(acsc(a + b*x)/x**3,x)