\(\int \frac {(a+b \text {csch}^{-1}(c x))^3}{x^2} \, dx\) [29]

Optimal result

Integrand size = 14, antiderivative size = 78 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^2} \, dx=6 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}-\frac {6 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x} \] Output:

6*b^3*c*(1+1/c^2/x^2)^(1/2)-6*b^2*(a+b*arccsch(c*x))/x+3*b*c*(1+1/c^2/x^2) 
^(1/2)*(a+b*arccsch(c*x))^2-(a+b*arccsch(c*x))^3/x
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {a^3+6 a b^2-3 a^2 b c \sqrt {1+\frac {1}{c^2 x^2}} x-6 b^3 c \sqrt {1+\frac {1}{c^2 x^2}} x+3 b \left (a^2+2 b^2-2 a b c \sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {csch}^{-1}(c x)+3 b^2 \left (a-b c \sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {csch}^{-1}(c x)^2+b^3 \text {csch}^{-1}(c x)^3}{x} \] Input:

Integrate[(a + b*ArcCsch[c*x])^3/x^2,x]
 

Output:

-((a^3 + 6*a*b^2 - 3*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 + 1/ 
(c^2*x^2)]*x + 3*b*(a^2 + 2*b^2 - 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch 
[c*x] + 3*b^2*(a - b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x]^2 + b^3*ArcCs 
ch[c*x]^3)/x)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6840, 3042, 3777, 26, 3042, 26, 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.

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

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

Input:

Int[(a + b*ArcCsch[c*x])^3/x^2,x]
 

Output:

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

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 3042

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

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 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]

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{3}}{x^{2}}d x\]

Input:

int((a+b*arccsch(c*x))^3/x^2,x)
 

Output:

int((a+b*arccsch(c*x))^3/x^2,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (74) = 148\).

Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.85 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \, {\left (b^{3} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \, {\left (2 \, a b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \] Input:

integrate((a+b*arccsch(c*x))^3/x^2,x, algorithm="fricas")
 

Output:

-(b^3*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))^3 - 3*(a^2*b + 2* 
b^3)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + a^3 + 6*a*b^2 - 3*(b^3*c*x*sqrt(( 
c^2*x^2 + 1)/(c^2*x^2)) - a*b^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 
1)/(c*x))^2 - 3*(2*a*b^2*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - a^2*b - 2*b^3 
)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)))/x
 

Sympy [F]

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

integrate((a+b*acsch(c*x))**3/x**2,x)
 

Output:

Integral((a + b*acsch(c*x))**3/x**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \operatorname {arcsch}\left (c x\right )^{3}}{x} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} a^{2} b + 6 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsch}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsch}\left (c x\right )^{2} + 2 \, c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {2 \, \operatorname {arcsch}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arcsch}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \] Input:

integrate((a+b*arccsch(c*x))^3/x^2,x, algorithm="maxima")
 

Output:

-b^3*arccsch(c*x)^3/x + 3*(c*sqrt(1/(c^2*x^2) + 1) - arccsch(c*x)/x)*a^2*b 
 + 6*(c*sqrt(1/(c^2*x^2) + 1)*arccsch(c*x) - 1/x)*a*b^2 + 3*(c*sqrt(1/(c^2 
*x^2) + 1)*arccsch(c*x)^2 + 2*c*sqrt(1/(c^2*x^2) + 1) - 2*arccsch(c*x)/x)* 
b^3 - 3*a*b^2*arccsch(c*x)^2/x - a^3/x
 

Giac [F]

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

integrate((a+b*arccsch(c*x))^3/x^2,x, algorithm="giac")
 

Output:

integrate((b*arccsch(c*x) + a)^3/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^2} \,d x \] Input:

int((a + b*asinh(1/(c*x)))^3/x^2,x)
 

Output:

int((a + b*asinh(1/(c*x)))^3/x^2, x)
 

Reduce [F]

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

int((a+b*acsch(c*x))^3/x^2,x)
 

Output:

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