3.1.37 \(\int \frac {1}{x^3 (a+b \text {csch}^{-1}(c x))} \, dx\) [37]

3.1.37.1 Optimal result

Integrand size = 14, antiderivative size = 63 \[ \int \frac {1}{x^3 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right )}{2 b} \]

output

-1/2*c^2*cosh(2*a/b)*Shi(2*a/b+2*arccsch(c*x))/b+1/2*c^2*Chi(2*a/b+2*arccs 
ch(c*x))*sinh(2*a/b)/b
 

3.1.37.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx=\frac {c^2 \left (\text {Chi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right )\right )}{2 b} \]

input

Integrate[1/(x^3*(a + b*ArcCsch[c*x])),x]
 

output

(c^2*(CoshIntegral[(2*a)/b + 2*ArcCsch[c*x]]*Sinh[(2*a)/b] - Cosh[(2*a)/b] 
*SinhIntegral[(2*a)/b + 2*ArcCsch[c*x]]))/(2*b)
 

3.1.37.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6840, 5971, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

\(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right )}{b}-\frac {i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right )}{b}\right )\)

input

Int[1/(x^3*(a + b*ArcCsch[c*x])),x]
 

output

(I/2)*c^2*(((-I)*CoshIntegral[(2*a)/b + 2*ArcCsch[c*x]]*Sinh[(2*a)/b])/b + 
 (I*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCsch[c*x]])/b)
 

3.1.37.3.1 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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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])
 

3.1.37.4 Maple [F]

\[\int \frac {1}{x^{3} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}d x\]

input

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

output

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

3.1.37.5 Fricas [F]

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

input

integrate(1/x^3/(a+b*arccsch(c*x)),x, algorithm="fricas")
 

output

integral(1/(b*x^3*arccsch(c*x) + a*x^3), x)
 

3.1.37.6 Sympy [F]

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

input

integrate(1/x**3/(a+b*acsch(c*x)),x)
 

output

Integral(1/(x**3*(a + b*acsch(c*x))), x)
 

3.1.37.7 Maxima [F]

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

input

integrate(1/x^3/(a+b*arccsch(c*x)),x, algorithm="maxima")
 

output

integrate(1/((b*arccsch(c*x) + a)*x^3), x)
 

3.1.37.8 Giac [F]

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

input

integrate(1/x^3/(a+b*arccsch(c*x)),x, algorithm="giac")
 

output

integrate(1/((b*arccsch(c*x) + a)*x^3), x)
 

3.1.37.9 Mupad [F(-1)]

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

input

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

output

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