\(\int \frac {(a+b \csc ^{-1}(c x))^3}{x^5} \, dx\) [32]

Optimal result

Integrand size = 14, antiderivative size = 208 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4} \] Output:

3/128*b^3*c*(1-1/c^2/x^2)^(1/2)/x^3+45/256*b^3*c^3*(1-1/c^2/x^2)^(1/2)/x-4 
5/256*b^3*c^4*arccsc(c*x)+3/32*b^2*(a+b*arccsc(c*x))/x^4+9/32*b^2*c^2*(a+b 
*arccsc(c*x))/x^2-3/16*b*c*(1-1/c^2/x^2)^(1/2)*(a+b*arccsc(c*x))^2/x^3-9/3 
2*b*c^3*(1-1/c^2/x^2)^(1/2)*(a+b*arccsc(c*x))^2/x+3/32*c^4*(a+b*arccsc(c*x 
))^3-1/4*(a+b*arccsc(c*x))^3/x^4
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {-64 a^3+24 a b^2-48 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+72 a b^2 c^2 x^2-72 a^2 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+24 b \left (-8 a^2+b^2 \left (1+3 c^2 x^2\right )-2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )\right ) \csc ^{-1}(c x)-24 b^2 \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )+a \left (8-3 c^4 x^4\right )\right ) \csc ^{-1}(c x)^2+8 b^3 \left (-8+3 c^4 x^4\right ) \csc ^{-1}(c x)^3+9 b \left (8 a^2-5 b^2\right ) c^4 x^4 \arcsin \left (\frac {1}{c x}\right )}{256 x^4} \] Input:

Integrate[(a + b*ArcCsc[c*x])^3/x^5,x]
 

Output:

(-64*a^3 + 24*a*b^2 - 48*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 6*b^3*c*Sqrt[1 
- 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2 - 72*a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x 
^3 + 45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 24*b*(-8*a^2 + b^2*(1 + 3*c^2* 
x^2) - 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcCsc[c*x] - 24*b 
^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2) + a*(8 - 3*c^4*x^4))*ArcCs 
c[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcCsc[c*x]^3 + 9*b*(8*a^2 - 5*b^2)*c^4* 
x^4*ArcSin[1/(c*x)])/(256*x^4)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.29, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5746, 4904, 3042, 3792, 3042, 3115, 3042, 3115, 24, 3792, 17, 3042, 3115, 24}

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.

\(\displaystyle \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx\)

\(\Big \downarrow \) 5746

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

\(\Big \downarrow \) 4904

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{c^2 x^2}d\csc ^{-1}(c x)-\frac {1}{8} b^2 \int \frac {1}{c^4 x^4}d\csc ^{-1}(c x)+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{8} b^2 \int \sin \left (\csc ^{-1}(c x)\right )^4d\csc ^{-1}(c x)+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{8} b^2 \left (\frac {3}{4} \int \frac {1}{c^2 x^2}d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{8} b^2 \left (\frac {3}{4} \int \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \int 1d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 24

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )\right )\)

\(\Big \downarrow \) 3792

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \int \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\csc ^{-1}(c x)+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )\right )\)

\(\Big \downarrow \) 17

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \int \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \left (\frac {1}{2} \int 1d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )\right )\)

\(\Big \downarrow \) 24

\(\displaystyle -c^4 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}-\frac {1}{2} b^2 \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )\right )+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{8 c^4 x^4}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c^3 x^3}-\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3 x^3}\right )\right )\right )\)

Input:

Int[(a + b*ArcCsc[c*x])^3/x^5,x]
 

Output:

-(c^4*((a + b*ArcCsc[c*x])^3/(4*c^4*x^4) - (3*b*(-1/8*(b^2*(-1/4*Sqrt[1 - 
1/(c^2*x^2)]/(c^3*x^3) + (3*(-1/2*Sqrt[1 - 1/(c^2*x^2)]/(c*x) + ArcCsc[c*x 
]/2))/4)) + (b*(a + b*ArcCsc[c*x]))/(8*c^4*x^4) - (Sqrt[1 - 1/(c^2*x^2)]*( 
a + b*ArcCsc[c*x])^2)/(4*c^3*x^3) + (3*(-1/2*(b^2*(-1/2*Sqrt[1 - 1/(c^2*x^ 
2)]/(c*x) + ArcCsc[c*x]/2)) + (b*(a + b*ArcCsc[c*x]))/(2*c^2*x^2) - (Sqrt[ 
1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(2*c*x) + (a + b*ArcCsc[c*x])^3/(6 
*b)))/4))/4))
 

Defintions of rubi rules used

rule 17

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

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

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

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 4904

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x 
_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) 
, x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 5746

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC 
sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n 
, 0] || LtQ[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(182)=364\).

Time = 1.16 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.30

method	result	size
parts	\(-\frac {a^{3}}{4 x^{4}}+b^{3} c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}+\frac {27 \,\operatorname {arccsc}\left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{32 c^{2} x^{2}}+\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arccsc}\left (c x \right )^{3}}{16}\right )-\frac {3 \,\operatorname {arccsc}\left (c x \right ) a^{2} b}{4 x^{4}}+\frac {9 a^{2} b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {9 a^{2} b c \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}-\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}+3 a \,b^{2} c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arccsc}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )\)	\(479\)
derivativedivides	\(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}+\frac {27 \,\operatorname {arccsc}\left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{32 c^{2} x^{2}}+\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arccsc}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arccsc}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arccsc}\left (c x \right )}{4 c^{4} x^{4}}+\frac {9 a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {9 a^{2} b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}-\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\)	\(485\)
default	\(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{32 c^{3} x^{3}}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \left (3 c^{2} x^{2}+2\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 c^{3} x^{3}}+\frac {27 \,\operatorname {arccsc}\left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{32 c^{2} x^{2}}+\frac {9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{64 c x}-\frac {3 \operatorname {arccsc}\left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arccsc}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arccsc}\left (c x \right )-3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {3 a^{2} b \,\operatorname {arccsc}\left (c x \right )}{4 c^{4} x^{4}}+\frac {9 a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {9 a^{2} b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}-\frac {3 a^{2} b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\)	\(485\)

Input:

int((a+b*arccsc(c*x))^3/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/4*a^3/x^4+b^3*c^4*(-1/4/c^4/x^4*arccsc(c*x)^3+3/32*arccsc(c*x)^2*(3*c^3 
*x^3*arccsc(c*x)-3*c^2*x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)-2*((c^2*x^2-1)/c^2/ 
x^2)^(1/2))/c^3/x^3+3/32*arccsc(c*x)/c^4/x^4+3/256*(3*c^2*x^2+2)/c^3/x^3*( 
(c^2*x^2-1)/c^2/x^2)^(1/2)+27/256*arccsc(c*x)-9/32*(c^2*x^2-1)/c^2/x^2*arc 
csc(c*x)+9/64/c/x*((c^2*x^2-1)/c^2/x^2)^(1/2)-3/16*arccsc(c*x)^3)-3/4*arcc 
sc(c*x)*a^2*b/x^4+9/32*a^2*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^( 
1/2)/x*arctan(1/(c^2*x^2-1)^(1/2))-9/32*a^2*b*c*(c^2*x^2-1)/((c^2*x^2-1)/c 
^2/x^2)^(1/2)/x^3-3/16*a^2*b/c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^5 
+3*a*b^2*c^4*(-1/4/c^4/x^4*arccsc(c*x)^2+1/16*arccsc(c*x)*(3*c^3*x^3*arccs 
c(c*x)-3*c^2*x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)-2*((c^2*x^2-1)/c^2/x^2)^(1/2) 
)/c^3/x^3-3/32*arccsc(c*x)^2+1/128*(3*c^2*x^2+2)^2/c^4/x^4)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{3} - 64 \, a^{3} + 24 \, a b^{2} + 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{4} x^{4} + 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b + 8 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{2} x^{2} + 16 \, a^{2} b - 2 \, b^{3} + 8 \, {\left (3 \, b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 16 \, {\left (3 \, a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{256 \, x^{4}} \] Input:

integrate((a+b*arccsc(c*x))^3/x^5,x, algorithm="fricas")
 

Output:

1/256*(72*a*b^2*c^2*x^2 + 8*(3*b^3*c^4*x^4 - 8*b^3)*arccsc(c*x)^3 - 64*a^3 
 + 24*a*b^2 + 24*(3*a*b^2*c^4*x^4 - 8*a*b^2)*arccsc(c*x)^2 + 3*(3*(8*a^2*b 
 - 5*b^3)*c^4*x^4 + 24*b^3*c^2*x^2 - 64*a^2*b + 8*b^3)*arccsc(c*x) - 3*(3* 
(8*a^2*b - 5*b^3)*c^2*x^2 + 16*a^2*b - 2*b^3 + 8*(3*b^3*c^2*x^2 + 2*b^3)*a 
rccsc(c*x)^2 + 16*(3*a*b^2*c^2*x^2 + 2*a*b^2)*arccsc(c*x))*sqrt(c^2*x^2 - 
1))/x^4
 

Sympy [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \] Input:

integrate((a+b*acsc(c*x))**3/x**5,x)
 

Output:

Integral((a + b*acsc(c*x))**3/x**5, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x^{5}} \,d x } \] Input:

integrate((a+b*arccsc(c*x))^3/x^5,x, algorithm="maxima")
 

Output:

-3/32*a^2*b*((3*c^5*arctan(c*x*sqrt(-1/(c^2*x^2) + 1)) + (3*c^8*x^3*(-1/(c 
^2*x^2) + 1)^(3/2) + 5*c^6*x*sqrt(-1/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) 
 - 1)^2 - 2*c^2*x^2*(1/(c^2*x^2) - 1) + 1))/c + 8*arccsc(c*x)/x^4) - 1/4*a 
^3/x^4 - 1/16*(4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*b^3*arc 
tan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 + 12*(2*(c^2*log(c*x + 
 1) + c^2*log(c*x - 1) - 2*c^2*log(x) + 1/x^2)*a*b^2*c^2*log(c)^2 + 64*b^3 
*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^7 - 
 x^5), x)*log(c)^2 - 64*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1) 
*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) + 128*b^3*c^2*int 
egrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^7 - 
x^5), x)*log(c) - 64*a*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2)/(c^2*x^7 - 
x^5), x)*log(c) + 128*a*b^2*c^2*integrate(1/16*x^2*log(x)/(c^2*x^7 - x^5), 
 x)*log(c) - 64*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c* 
x - 1)))*log(c^2*x^2)*log(x)/(c^2*x^7 - x^5), x) + 64*b^3*c^2*integrate(1/ 
16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^7 - x^5), x 
) - 64*a*b^2*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)) 
)^2/(c^2*x^7 - x^5), x) + 16*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x 
 + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^7 - x^5), x) + 16*a*b^2*c^2*inte 
grate(1/16*x^2*log(c^2*x^2)^2/(c^2*x^7 - x^5), x) - 64*a*b^2*c^2*integrate 
(1/16*x^2*log(c^2*x^2)*log(x)/(c^2*x^7 - x^5), x) + 64*a*b^2*c^2*integr...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (182) = 364\).

Time = 0.15 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((a+b*arccsc(c*x))^3/x^5,x, algorithm="giac")
 

Output:

-1/256*(64*b^3*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))^3 + 192*a*b^2*c^3*( 
1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))^2 + 128*b^3*c^3*(1/(c^2*x^2) - 1)*arcsi 
n(1/(c*x))^3 + 192*a^2*b*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x)) - 24*b^3* 
c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x)) + 384*a*b^2*c^3*(1/(c^2*x^2) - 1)* 
arcsin(1/(c*x))^2 + 40*b^3*c^3*arcsin(1/(c*x))^3 - 24*a*b^2*c^3*(1/(c^2*x^ 
2) - 1)^2 + 384*a^2*b*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) - 120*b^3*c^3* 
(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) + 120*a*b^2*c^3*arcsin(1/(c*x))^2 - 48*b 
^3*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x))^2/x - 120*a*b^2*c^3*(1/(c^ 
2*x^2) - 1) + 120*a^2*b*c^3*arcsin(1/(c*x)) - 51*b^3*c^3*arcsin(1/(c*x)) - 
 96*a*b^2*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x))/x + 120*b^3*c^2*sqr 
t(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))^2/x - 51*a*b^2*c^3 - 48*a^2*b*c^2*(-1/ 
(c^2*x^2) + 1)^(3/2)/x + 6*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2)/x + 240*a*b^2* 
c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))/x + 120*a^2*b*c^2*sqrt(-1/(c^2* 
x^2) + 1)/x - 51*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)/x + 64*a^3/(c*x^4))*c
 

Mupad [F(-1)]

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

int((a + b*asin(1/(c*x)))^3/x^5,x)
                                                                                    
                                                                                    
 

Output:

int((a + b*asin(1/(c*x)))^3/x^5, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx=\frac {12 \left (\int \frac {\mathit {acsc} \left (c x \right )}{x^{5}}d x \right ) a^{2} b \,x^{4}+4 \left (\int \frac {\mathit {acsc} \left (c x \right )^{3}}{x^{5}}d x \right ) b^{3} x^{4}+12 \left (\int \frac {\mathit {acsc} \left (c x \right )^{2}}{x^{5}}d x \right ) a \,b^{2} x^{4}-a^{3}}{4 x^{4}} \] Input:

int((a+b*acsc(c*x))^3/x^5,x)
 

Output:

(12*int(acsc(c*x)/x**5,x)*a**2*b*x**4 + 4*int(acsc(c*x)**3/x**5,x)*b**3*x* 
*4 + 12*int(acsc(c*x)**2/x**5,x)*a*b**2*x**4 - a**3)/(4*x**4)