\(\int \frac {1}{x^4 (a+b \text {csch}^{-1}(c x))} \, dx\) [38]

Optimal result

Integrand size = 14, antiderivative size = 117 \[ \int \frac {1}{x^4 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {csch}^{-1}(c x)\right )}{4 b}-\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{4 b}+\frac {c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {csch}^{-1}(c x)\right )}{4 b} \] Output:

1/4*c^3*cosh(a/b)*Chi(a/b+arccsch(c*x))/b-1/4*c^3*cosh(3*a/b)*Chi(3*a/b+3* 
arccsch(c*x))/b-1/4*c^3*sinh(a/b)*Shi(a/b+arccsch(c*x))/b+1/4*c^3*sinh(3*a 
/b)*Shi(3*a/b+3*arccsch(c*x))/b
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=-\frac {c^3 \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )\right )\right )}{4 b} \] Input:

Integrate[1/(x^4*(a + b*ArcCsch[c*x])),x]
 

Output:

-1/4*(c^3*(-(Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]]) + Cosh[(3*a)/b]*C 
oshIntegral[3*(a/b + ArcCsch[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcCsch 
[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCsch[c*x])]))/b
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6840, 5971, 2009}

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.

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

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

\(\displaystyle -c^3 \left (-\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{4 b}+\frac {\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {csch}^{-1}(c x)\right )}{4 b}+\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {csch}^{-1}(c x)\right )}{4 b}-\frac {\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {csch}^{-1}(c x)\right )}{4 b}\right )\)

Input:

Int[1/(x^4*(a + b*ArcCsch[c*x])),x]
 

Output:

-(c^3*(-1/4*(Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]])/b + (Cosh[(3*a)/b 
]*CoshIntegral[(3*a)/b + 3*ArcCsch[c*x]])/(4*b) + (Sinh[a/b]*SinhIntegral[ 
a/b + ArcCsch[c*x]])/(4*b) - (Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCs 
ch[c*x]])/(4*b)))
 

Defintions of rubi rules used

rule 2009

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

rule 5971

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

\[\int \frac {1}{x^{4} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate(1/x^4/(a+b*arccsch(c*x)),x, algorithm="fricas")
 

Output:

integral(1/(b*x^4*arccsch(c*x) + a*x^4), x)
 

Sympy [F]

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

integrate(1/x**4/(a+b*acsch(c*x)),x)
 

Output:

Integral(1/(x**4*(a + b*acsch(c*x))), x)
 

Maxima [F]

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

integrate(1/x^4/(a+b*arccsch(c*x)),x, algorithm="maxima")
 

Output:

integrate(1/((b*arccsch(c*x) + a)*x^4), x)
 

Giac [F]

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

integrate(1/x^4/(a+b*arccsch(c*x)),x, algorithm="giac")
 

Output:

integrate(1/((b*arccsch(c*x) + a)*x^4), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(1/(acsch(c*x)*b*x**4 + a*x**4),x)