3.1.25 \(\int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx\) [25]

3.1.25.1 Optimal result

Integrand size = 10, antiderivative size = 180 \[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}+\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \csc ^{-1}(a+b x)}{3 a^3}-\frac {\csc ^{-1}(a+b x)}{3 x^3}-\frac {\left (2-5 a^2+6 a^4\right ) b^3 \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}} \]

output

-1/3*b^3*arccsc(b*x+a)/a^3-1/3*arccsc(b*x+a)/x^3-1/3*(6*a^4-5*a^2+2)*b^3*a 
rctan((a-tan(1/2*arccsc(b*x+a)))/(-a^2+1)^(1/2))/a^3/(-a^2+1)^(5/2)-1/6*b* 
(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/a/(-a^2+1)/x^2+1/6*(-5*a^2+2)*b^2*(b*x+a)*(1 
-1/(b*x+a)^2)^(1/2)/a^2/(-a^2+1)^2/x
 

3.1.25.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.34 \[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=\frac {1}{6} \left (\frac {b \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (a^4+a b x-4 a^3 b x+2 b^2 x^2-a^2 \left (1+5 b^2 x^2\right )\right )}{a^2 \left (-1+a^2\right )^2 x^2}-\frac {2 \csc ^{-1}(a+b x)}{x^3}-\frac {2 b^3 \arcsin \left (\frac {1}{a+b x}\right )}{a^3}+\frac {i \left (2-5 a^2+6 a^4\right ) b^3 \log \left (\frac {12 a^3 \left (-1+a^2\right )^2 \left (-\frac {i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2}}-(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{\left (2-5 a^2+6 a^4\right ) b^3 x}\right )}{a^3 \left (1-a^2\right )^{5/2}}\right ) \]

input

Integrate[ArcCsc[a + b*x]/x^4,x]
 

output

((b*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(a^4 + a*b*x - 4*a^3* 
b*x + 2*b^2*x^2 - a^2*(1 + 5*b^2*x^2)))/(a^2*(-1 + a^2)^2*x^2) - (2*ArcCsc 
[a + b*x])/x^3 - (2*b^3*ArcSin[(a + b*x)^(-1)])/a^3 + (I*(2 - 5*a^2 + 6*a^ 
4)*b^3*Log[(12*a^3*(-1 + a^2)^2*(((-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]))/((2 - 5*a^2 
+ 6*a^4)*b^3*x)])/(a^3*(1 - a^2)^(5/2)))/6
 

3.1.25.3 Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {5782, 4927, 3042, 4272, 3042, 4548, 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.

\(\displaystyle \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx\)

\(\Big \downarrow \) 5782

\(\displaystyle -b^3 \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4 x^4}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 4927

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\int -\frac {2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) (a+b x)}{b x}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\int \frac {2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) \csc \left (\csc ^{-1}(a+b x)\right )}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 4407

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {\left (6 a^4-5 a^2+2\right ) \int -\frac {a+b x}{b x}d\csc ^{-1}(a+b x)}{a}+\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {\left (6 a^4-5 a^2+2\right ) \int \frac {\csc \left (\csc ^{-1}(a+b x)\right )}{a-\csc \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}+\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 4318

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}-\frac {\left (6 a^4-5 a^2+2\right ) \int \frac {1}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}-\frac {\left (6 a^4-5 a^2+2\right ) \int \frac {1}{1-a \sin \left (\csc ^{-1}(a+b x)\right )}d\csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 3139

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}-\frac {2 \left (6 a^4-5 a^2+2\right ) \int \frac {1}{\tan ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+1}d\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\frac {\frac {4 \left (6 a^4-5 a^2+2\right ) \int \frac {1}{-\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )^2-4 \left (1-a^2\right )}d\left (2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a\right )}{a}+\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}+\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -b^3 \left (\frac {1}{3} \left (\frac {\sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)}{2 a \left (1-a^2\right ) b^2 x^2}+\frac {\frac {\frac {2 \left (1-a^2\right )^2 \csc ^{-1}(a+b x)}{a}-\frac {2 \left (6 a^4-5 a^2+2\right ) \arctan \left (\frac {2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 a}{2 \sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}}{a \left (1-a^2\right )}-\frac {\left (2-5 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{a \left (1-a^2\right ) b x}}{2 a \left (1-a^2\right )}\right )+\frac {\csc ^{-1}(a+b x)}{3 b^3 x^3}\right )\)

input

Int[ArcCsc[a + b*x]/x^4,x]
 

output

-(b^3*(ArcCsc[a + b*x]/(3*b^3*x^3) + (((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) 
/(2*a*(1 - a^2)*b^2*x^2) + (-(((2 - 5*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(- 
2)])/(a*(1 - a^2)*b*x)) + ((2*(1 - a^2)^2*ArcCsc[a + b*x])/a - (2*(2 - 5*a 
^2 + 6*a^4)*ArcTan[(-2*a + 2*Tan[ArcCsc[a + b*x]/2])/(2*Sqrt[1 - a^2])])/( 
a*Sqrt[1 - a^2]))/(a*(1 - a^2)))/(2*a*(1 - a^2)))/3))
 

3.1.25.3.1 Defintions of rubi rules used

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 4548

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - 
a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 
 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[e + f*x])^( 
m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x 
] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, 
 b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
 

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]
 

3.1.25.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(160)=320\).

Time = 0.72 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.93

method	result	size
parts	\(-\frac {\operatorname {arccsc}\left (b x +a \right )}{3 x^{3}}-\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (2 \left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{4} b^{2} x^{2}-6 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{6} b^{2} x^{2}-4 \left (a^{2}-1\right )^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) a^{2} b^{2} x^{2}+5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{3} b x +11 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{4} b^{2} x^{2}+2 b^{2} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) x^{2} \left (a^{2}-1\right )^{\frac {3}{2}}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{4}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a b x -7 \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{2} b^{2} x^{2}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{\frac {3}{2}} a^{2}+2 b^{2} \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) x^{2}\right )}{6 \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{3} \left (a^{2}-1\right )^{\frac {7}{2}} x^{2}}\)	\(528\)
derivativedivides	\(\text {Expression too large to display}\)	\(970\)
default	\(\text {Expression too large to display}\)	\(970\)

input

int(arccsc(b*x+a)/x^4,x,method=_RETURNVERBOSE)
 

output

-1/3*arccsc(b*x+a)/x^3-1/6*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*(2*(a^2-1)^(3/2 
)*arctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*a^4*b^2*x^2-6*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^6*b^2*x^2-4*(a^2-1)^(3/2 
)*arctan(1/(b^2*x^2+2*a*b*x+a^2-1)^(1/2))*a^2*b^2*x^2+5*(b^2*x^2+2*a*b*x+a 
^2-1)^(1/2)*(a^2-1)^(3/2)*a^3*b*x+11*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^2*x^2+2*b^2*arctan(1/(b^2*x^2+2*a*b*x+a 
^2-1)^(1/2))*x^2*(a^2-1)^(3/2)-(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*(a^2-1)^(3/2) 
*a^4-2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*(a^2-1)^(3/2)*a*b*x-7*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^2*x^2+(b^2*x^2+2* 
a*b*x+a^2-1)^(1/2)*(a^2-1)^(3/2)*a^2+2*b^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)*x^2)/((b^2*x^2+2*a*b*x+a^2-1)/(b*x+a)^2 
)^(1/2)/(b*x+a)/a^3/(a^2-1)^(7/2)/x^2
 

3.1.25.5 Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.04 \[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=\left [\frac {{\left (6 \, a^{4} - 5 \, a^{2} + 2\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) + 4 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{3} x^{3} - 2 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \operatorname {arccsc}\left (b x + a\right ) - {\left ({\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{2} x^{2} - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}}, -\frac {2 \, {\left (6 \, a^{4} - 5 \, a^{2} + 2\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - 4 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{3} x^{3} + 2 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \operatorname {arccsc}\left (b x + a\right ) + {\left ({\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{2} x^{2} - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}}\right ] \]

input

integrate(arccsc(b*x+a)/x^4,x, algorithm="fricas")
 

output

[1/6*((6*a^4 - 5*a^2 + 2)*sqrt(a^2 - 1)*b^3*x^3*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) + 4*(a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*arctan(-b*x 
 - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (5*a^5 - 7*a^3 + 2*a)*b^3*x^3 
- 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*arccsc(b*x + a) - ((5*a^5 - 7*a^3 + 2*a)*b 
^2*x^2 - (a^6 - 2*a^4 + a^2)*b*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/((a^9 
 - 3*a^7 + 3*a^5 - a^3)*x^3), -1/6*(2*(6*a^4 - 5*a^2 + 2)*sqrt(-a^2 + 1)*b 
^3*x^3*arctan(-(sqrt(-a^2 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*sqr 
t(-a^2 + 1))/(a^2 - 1)) - 4*(a^6 - 3*a^4 + 3*a^2 - 1)*b^3*x^3*arctan(-b*x 
- a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (5*a^5 - 7*a^3 + 2*a)*b^3*x^3 + 
 2*(a^9 - 3*a^7 + 3*a^5 - a^3)*arccsc(b*x + a) + ((5*a^5 - 7*a^3 + 2*a)*b^ 
2*x^2 - (a^6 - 2*a^4 + a^2)*b*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/((a^9 
- 3*a^7 + 3*a^5 - a^3)*x^3)]
 

3.1.25.6 Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{4}}\, dx \]

input

integrate(acsc(b*x+a)/x**4,x)
 

output

Integral(acsc(a + b*x)/x**4, x)
 

3.1.25.7 Maxima [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x^{4}} \,d x } \]

input

integrate(arccsc(b*x+a)/x^4,x, algorithm="maxima")
 

output

-1/3*(3*x^3*integrate(1/3*(b^2*x + a*b)*e^(1/2*log(b*x + a + 1) + 1/2*log( 
b*x + a - 1))/(b^2*x^5 + 2*a*b*x^4 + (a^2 - 1)*x^3 + (b^2*x^5 + 2*a*b*x^4 
+ (a^2 - 1)*x^3)*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^3
 

3.1.25.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (156) = 312\).

Time = 0.37 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.50 \[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=-\frac {1}{3} \, b {\left (\frac {{\left (6 \, a^{4} b^{2} - 5 \, a^{2} b^{2} + 2 \, b^{2}\right )} \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{7} - 2 \, a^{5} + a^{3}\right )} \sqrt {-a^{2} + 1}} + \frac {4 \, {\left (b x + a\right )}^{3} a^{3} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 10 \, {\left (b x + a\right )}^{2} a^{4} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} - {\left (b x + a\right )}^{3} a b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + {\left (b x + a\right )}^{2} a^{2} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 16 \, {\left (b x + a\right )} a^{3} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} - 2 \, {\left (b x + a\right )}^{2} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} - 7 \, {\left (b x + a\right )} a b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 5 \, a^{2} b^{2} - 2 \, b^{2}}{{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} {\left ({\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 2 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1\right )}^{2}} + \frac {{\left (\frac {3 \, a b^{2}}{b x + a} - \frac {3 \, a^{2} b^{2}}{{\left (b x + a\right )}^{2}} - b^{2}\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a^{3} {\left (\frac {a}{b x + a} - 1\right )}^{3}}\right )} \]

input

integrate(arccsc(b*x+a)/x^4,x, algorithm="giac")
 

output

-1/3*b*((6*a^4*b^2 - 5*a^2*b^2 + 2*b^2)*arctan(((b*x + a)*(sqrt(-1/(b*x + 
a)^2 + 1) - 1) + a)/sqrt(-a^2 + 1))/((a^7 - 2*a^5 + a^3)*sqrt(-a^2 + 1)) + 
 (4*(b*x + a)^3*a^3*b^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 10*(b*x + a)^2* 
a^4*b^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 - (b*x + a)^3*a*b^2*(sqrt(-1/(b*x 
 + a)^2 + 1) - 1)^3 + (b*x + a)^2*a^2*b^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 
 + 16*(b*x + a)*a^3*b^2*(sqrt(-1/(b*x + a)^2 + 1) - 1) - 2*(b*x + a)^2*b^2 
*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 - 7*(b*x + a)*a*b^2*(sqrt(-1/(b*x + a)^2 
 + 1) - 1) + 5*a^2*b^2 - 2*b^2)/((a^6 - 2*a^4 + a^2)*((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)^2) + (3*a*b^2/(b*x + a) - 3*a^2*b^2/(b*x + a)^2 - b^2)*arcsin(-1/((b*x 
+ a)*(a/(b*x + a) - 1) - a))/(a^3*(a/(b*x + a) - 1)^3))
 

3.1.25.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^4} \,d x \]

input

int(asin(1/(a + b*x))/x^4,x)
 

output

int(asin(1/(a + b*x))/x^4, x)