\(\int \frac {(a+b \text {csch}^{-1}(c x))^3}{x^4} \, dx\) [31]

Optimal result

Integrand size = 14, antiderivative size = 166 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=-\frac {14}{9} b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3} \] Output:

-14/9*b^3*c^3*(1+1/c^2/x^2)^(1/2)+2/27*b^3*c^3*(1+1/c^2/x^2)^(3/2)-2/9*b^2 
*(a+b*arccsch(c*x))/x^3+4/3*b^2*c^2*(a+b*arccsch(c*x))/x-2/3*b*c^3*(1+1/c^ 
2/x^2)^(1/2)*(a+b*arccsch(c*x))^2+1/3*b*c*(1+1/c^2/x^2)^(1/2)*(a+b*arccsch 
(c*x))^2/x^2-1/3*(a+b*arccsch(c*x))^3/x^3
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\frac {-9 a^3+2 b^3 c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-20 c^2 x^2\right )+9 a^2 b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-2 c^2 x^2\right )+6 a b^2 \left (-1+6 c^2 x^2\right )+3 b \left (-9 a^2+6 a b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-2 c^2 x^2\right )+2 b^2 \left (-1+6 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)-9 b^2 \left (3 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-1+2 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)^2-9 b^3 \text {csch}^{-1}(c x)^3}{27 x^3} \] Input:

Integrate[(a + b*ArcCsch[c*x])^3/x^4,x]
 

Output:

(-9*a^3 + 2*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 20*c^2*x^2) + 9*a^2*b*c*Sqr 
t[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2) + 6*a*b^2*(-1 + 6*c^2*x^2) + 3*b*(-9* 
a^2 + 6*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2) + 2*b^2*(-1 + 6*c^2* 
x^2))*ArcCsch[c*x] - 9*b^2*(3*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-1 + 2*c^2* 
x^2))*ArcCsch[c*x]^2 - 9*b^3*ArcCsch[c*x]^3)/(27*x^3)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.19, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {6840, 5969, 3042, 26, 3792, 26, 3042, 26, 3113, 2009, 3777, 3042, 3777, 26, 3042, 26, 3118}

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.

\(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx\)

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3792

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \int \frac {i \left (a+b \text {csch}^{-1}(c x)\right )^2}{c x}d\text {csch}^{-1}(c x)+\frac {2}{9} b^2 \int -\frac {i}{c^3 x^3}d\text {csch}^{-1}(c x)+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} i \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{c x}d\text {csch}^{-1}(c x)-\frac {2}{9} i b^2 \int \frac {1}{c^3 x^3}d\text {csch}^{-1}(c x)+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} i \int -i \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {2}{9} i b^2 \int i \sin \left (i \text {csch}^{-1}(c x)\right )^3d\text {csch}^{-1}(c x)+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)+\frac {2}{9} b^2 \int \sin \left (i \text {csch}^{-1}(c x)\right )^3d\text {csch}^{-1}(c x)+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\)

\(\Big \downarrow \) 3113

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)+\frac {2}{9} i b^2 \int -\frac {1}{c^2 x^2}d\sqrt {1+\frac {1}{c^2 x^2}}+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \int \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)\right )+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(c x)\right )+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \left (\frac {a+b \text {csch}^{-1}(c x)}{c x}-i b \int -\frac {i}{c x}d\text {csch}^{-1}(c x)\right )\right )+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \left (\frac {a+b \text {csch}^{-1}(c x)}{c x}-b \int \frac {1}{c x}d\text {csch}^{-1}(c x)\right )\right )+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \left (\frac {a+b \text {csch}^{-1}(c x)}{c x}-b \int -i \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)\right )\right )+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \left (\frac {a+b \text {csch}^{-1}(c x)}{c x}+i b \int \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)\right )\right )+\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 c^3 x^3}-i b \left (\frac {2 i b \left (a+b \text {csch}^{-1}(c x)\right )}{9 c^3 x^3}-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-2 i b \left (\frac {a+b \text {csch}^{-1}(c x)}{c x}-b \sqrt {\frac {1}{c^2 x^2}+1}\right )\right )+\frac {2}{9} i b^2 \left (\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{3} \left (\frac {1}{c^2 x^2}+1\right )^{3/2}\right )\right )\right )\)

Input:

Int[(a + b*ArcCsch[c*x])^3/x^4,x]
 

Output:

-(c^3*((a + b*ArcCsch[c*x])^3/(3*c^3*x^3) - I*b*(((2*I)/9)*b^2*(Sqrt[1 + 1 
/(c^2*x^2)] - (1 + 1/(c^2*x^2))^(3/2)/3) + (((2*I)/9)*b*(a + b*ArcCsch[c*x 
]))/(c^3*x^3) - ((I/3)*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(c^2* 
x^2) + (2*(I*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2 - (2*I)*b*(-(b*S 
qrt[1 + 1/(c^2*x^2)]) + (a + b*ArcCsch[c*x])/(c*x))))/3)))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

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 5969

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

rule 6840

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, A 
rcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{3}}{x^{4}}d x\]

Input:

int((a+b*arccsch(c*x))^3/x^4,x)
 

Output:

int((a+b*arccsch(c*x))^3/x^4,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (144) = 288\).

Time = 0.10 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 9 \, a^{3} - 6 \, a b^{2} - 9 \, {\left (3 \, a b^{2} + {\left (2 \, b^{3} c^{3} x^{3} - b^{3} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b - 2 \, b^{3} - 6 \, {\left (2 \, a b^{2} c^{3} x^{3} - a b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (9 \, a^{2} b + 20 \, b^{3}\right )} c^{3} x^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{27 \, x^{3}} \] Input:

integrate((a+b*arccsch(c*x))^3/x^4,x, algorithm="fricas")
 

Output:

1/27*(36*a*b^2*c^2*x^2 - 9*b^3*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1) 
/(c*x))^3 - 9*a^3 - 6*a*b^2 - 9*(3*a*b^2 + (2*b^3*c^3*x^3 - b^3*c*x)*sqrt( 
(c^2*x^2 + 1)/(c^2*x^2)))*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x 
))^2 + 3*(12*b^3*c^2*x^2 - 9*a^2*b - 2*b^3 - 6*(2*a*b^2*c^3*x^3 - a*b^2*c* 
x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 
 1)/(c*x)) - (2*(9*a^2*b + 20*b^3)*c^3*x^3 - (9*a^2*b + 2*b^3)*c*x)*sqrt(( 
c^2*x^2 + 1)/(c^2*x^2)))/x^3
 

Sympy [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \] Input:

integrate((a+b*acsch(c*x))**3/x**4,x)
 

Output:

Integral((a + b*acsch(c*x))**3/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

integrate((a+b*arccsch(c*x))^3/x^4,x, algorithm="maxima")
 

Output:

1/3*a^2*b*((c^4*(1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 
 3*arccsch(c*x)/x^3) - 1/3*b^3*log(sqrt(c^2*x^2 + 1) + 1)^3/x^3 - 1/3*a^3/ 
x^3 - integrate((b^3*log(c)^3 - 3*a*b^2*log(c)^2 + (b^3*c^2*x^2 + b^3)*log 
(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c) - a* 
b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x)^2 + (3*b^3*log(c) - 3*a*b^2 
 + 3*(b^3*c^2*log(c) - a*b^2*c^2)*x^2 + 3*(b^3*c^2*x^2 + b^3)*log(x) + sqr 
t(c^2*x^2 + 1)*(3*b^3*log(c) - 3*a*b^2 + (b^3*c^2*(3*log(c) - 1) - 3*a*b^2 
*c^2)*x^2 + 3*(b^3*c^2*x^2 + b^3)*log(x)))*log(sqrt(c^2*x^2 + 1) + 1)^2 + 
3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c)) 
*x^2)*log(x) - 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a* 
b^2*c^2*log(c))*x^2 + (b^3*c^2*x^2 + b^3)*log(x)^2 + 2*(b^3*log(c) - a*b^2 
 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x) + (b^3*log(c)^2 - 2*a*b^2*log( 
c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2 + (b^3*c^2*x^2 + b^3)*log 
(x)^2 + 2*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x))* 
sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 1) + (b^3*log(c)^3 - 3*a*b^2*lo 
g(c)^2 + (b^3*c^2*x^2 + b^3)*log(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*lo 
g(c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*lo 
g(x)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^ 
2*log(c))*x^2)*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^6 + x^4 + (c^2*x^6 + x^4) 
*sqrt(c^2*x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

integrate((a+b*arccsch(c*x))^3/x^4,x, algorithm="giac")
 

Output:

integrate((b*arccsch(c*x) + a)^3/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \] Input:

int((a + b*asinh(1/(c*x)))^3/x^4,x)
 

Output:

int((a + b*asinh(1/(c*x)))^3/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx=\frac {9 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{4}}d x \right ) a^{2} b \,x^{3}+3 \left (\int \frac {\mathit {acsch} \left (c x \right )^{3}}{x^{4}}d x \right ) b^{3} x^{3}+9 \left (\int \frac {\mathit {acsch} \left (c x \right )^{2}}{x^{4}}d x \right ) a \,b^{2} x^{3}-a^{3}}{3 x^{3}} \] Input:

int((a+b*acsch(c*x))^3/x^4,x)
 

Output:

(9*int(acsch(c*x)/x**4,x)*a**2*b*x**3 + 3*int(acsch(c*x)**3/x**4,x)*b**3*x 
**3 + 9*int(acsch(c*x)**2/x**4,x)*a*b**2*x**3 - a**3)/(3*x**3)