\(\int \frac {(a+b \text {csch}^{-1}(c x))^2}{x^5} \, dx\) [23]

Optimal result

Integrand size = 14, antiderivative size = 120 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=-\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{16 x}+\frac {3}{32} c^4 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4} \] Output:

-1/32*b^2/x^4+3/32*b^2*c^2/x^2+1/8*b*c*(1+1/c^2/x^2)^(1/2)*(a+b*arccsch(c* 
x))/x^3-3/16*b*c^3*(1+1/c^2/x^2)^(1/2)*(a+b*arccsch(c*x))/x+3/32*c^4*(a+b* 
arccsch(c*x))^2-1/4*(a+b*arccsch(c*x))^2/x^4
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\frac {-8 a^2-b^2+4 a b c \sqrt {1+\frac {1}{c^2 x^2}} x+3 b^2 c^2 x^2-6 a b c^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3-2 b \left (8 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-2+3 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)+b^2 \left (-8+3 c^4 x^4\right ) \text {csch}^{-1}(c x)^2+6 a b c^4 x^4 \text {arcsinh}\left (\frac {1}{c x}\right )}{32 x^4} \] Input:

Integrate[(a + b*ArcCsch[c*x])^2/x^5,x]
 

Output:

(-8*a^2 - b^2 + 4*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + 3*b^2*c^2*x^2 - 6*a*b*c^ 
3*Sqrt[1 + 1/(c^2*x^2)]*x^3 - 2*b*(8*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-2 + 
 3*c^2*x^2))*ArcCsch[c*x] + b^2*(-8 + 3*c^4*x^4)*ArcCsch[c*x]^2 + 6*a*b*c^ 
4*x^4*ArcSinh[1/(c*x)])/(32*x^4)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6840, 5969, 3042, 3791, 25, 3042, 25, 3791, 17}

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.

\(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx\)

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \int -\frac {a+b \text {csch}^{-1}(c x)}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {a+b \text {csch}^{-1}(c x)}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (-\frac {3}{4} \int -\left (\left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^2\right )d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \left (\frac {1}{2} \int \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c x}+\frac {b}{4 c^2 x^2}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\)

\(\Big \downarrow \) 17

\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \left (-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c x}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 b}+\frac {b}{4 c^2 x^2}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\)

Input:

Int[(a + b*ArcCsch[c*x])^2/x^5,x]
 

Output:

-(c^4*((a + b*ArcCsch[c*x])^2/(4*c^4*x^4) - (b*(-1/16*b/(c^4*x^4) + (Sqrt[ 
1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(4*c^3*x^3) + (3*(b/(4*c^2*x^2) - ( 
Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(2*c*x) + (a + b*ArcCsch[c*x]) 
^2/(4*b)))/4))/2))
 

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 25

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

rule 3042

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

rule 3791

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

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{2}}{x^{5}}d x\]

Input:

int((a+b*arccsch(c*x))^2/x^5,x)
 

Output:

int((a+b*arccsch(c*x))^2/x^5,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\frac {3 \, b^{2} c^{2} x^{2} + {\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 8 \, a^{2} - b^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} - 8 \, a b - {\left (3 \, b^{2} c^{3} x^{3} - 2 \, b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (3 \, a b c^{3} x^{3} - 2 \, a b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{32 \, x^{4}} \] Input:

integrate((a+b*arccsch(c*x))^2/x^5,x, algorithm="fricas")
 

Output:

1/32*(3*b^2*c^2*x^2 + (3*b^2*c^4*x^4 - 8*b^2)*log((c*x*sqrt((c^2*x^2 + 1)/ 
(c^2*x^2)) + 1)/(c*x))^2 - 8*a^2 - b^2 + 2*(3*a*b*c^4*x^4 - 8*a*b - (3*b^2 
*c^3*x^3 - 2*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*(3*a*b*c^3*x^3 - 2*a*b*c*x)*sqrt((c^2*x^ 
2 + 1)/(c^2*x^2)))/x^4
 

Sympy [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{5}}\, dx \] Input:

integrate((a+b*acsch(c*x))**2/x**5,x)
 

Output:

Integral((a + b*acsch(c*x))**2/x**5, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((a+b*arccsch(c*x))^2/x^5,x, algorithm="maxima")
 

Output:

1/32*a*b*((3*c^5*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) - 3*c^5*log(c*x*sqrt(1 
/(c^2*x^2) + 1) - 1) - 2*(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 - 16*arccsch(c*x)/x^4) - 1/4*b^2*(log(sqrt(c^2*x^2 + 1) + 1)^2 
/x^4 + 4*integrate(-1/2*(2*c^2*x^2*log(c)^2 + 2*(c^2*x^2 + 1)*log(x)^2 + 2 
*log(c)^2 + 4*(c^2*x^2*log(c) + log(c))*log(x) - (4*c^2*x^2*log(c) + 4*(c^ 
2*x^2 + 1)*log(x) + (c^2*x^2*(4*log(c) - 1) + 4*(c^2*x^2 + 1)*log(x) + 4*l 
og(c))*sqrt(c^2*x^2 + 1) + 4*log(c))*log(sqrt(c^2*x^2 + 1) + 1) + 2*(c^2*x 
^2*log(c)^2 + (c^2*x^2 + 1)*log(x)^2 + log(c)^2 + 2*(c^2*x^2*log(c) + log( 
c))*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^7 + x^5 + (c^2*x^7 + x^5)*sqrt(c^2*x 
^2 + 1)), x)) - 1/4*a^2/x^4
 

Giac [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((a+b*arccsch(c*x))^2/x^5,x, algorithm="giac")
 

Output:

integrate((b*arccsch(c*x) + a)^2/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^5} \,d x \] Input:

int((a + b*asinh(1/(c*x)))^2/x^5,x)
 

Output:

int((a + b*asinh(1/(c*x)))^2/x^5, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\frac {8 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{5}}d x \right ) a b \,x^{4}+4 \left (\int \frac {\mathit {acsch} \left (c x \right )^{2}}{x^{5}}d x \right ) b^{2} x^{4}-a^{2}}{4 x^{4}} \] Input:

int((a+b*acsch(c*x))^2/x^5,x)
                                                                                    
                                                                                    
 

Output:

(8*int(acsch(c*x)/x**5,x)*a*b*x**4 + 4*int(acsch(c*x)**2/x**5,x)*b**2*x**4 
 - a**2)/(4*x**4)