\(\int \frac {a+b \text {csch}^{-1}(c x)}{x} \, dx\) [8]

Optimal result

Integrand size = 12, antiderivative size = 56 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x} \, dx=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \] Output:

1/2*(a+b*arccsch(c*x))^2/b-(a+b*arccsch(c*x))*ln(1-(1/c/x+(1+1/c^2/x^2)^(1 
/2))^2)-1/2*b*polylog(2,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x} \, dx=\frac {1}{2} b \text {csch}^{-1}(c x)^2-b \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+a \log (x)-\frac {1}{2} b \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \] Input:

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

Output:

(b*ArcCsch[c*x]^2)/2 - b*ArcCsch[c*x]*Log[1 - E^(2*ArcCsch[c*x])] + a*Log[ 
x] - (b*PolyLog[2, E^(2*ArcCsch[c*x])])/2
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.73, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6836, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6836

\(\displaystyle -\int x \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )d\frac {1}{x}\)

\(\Big \downarrow \) 6190

\(\displaystyle -\frac {\int -\left (\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{b}\right )\right )d\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{b}\right )d\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \tan \left (\frac {i a}{b}-\frac {i \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}+\frac {\pi }{2}\right )d\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}\right )d\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}-i \pi }\right )d\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1+e^{-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {i}{2 x^2}\right )}{b}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int x \log \left (1+e^{\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}-i \pi }-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1+e^{-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {i}{2 x^2}\right )}{b}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}\left (2,-a-b \text {arcsinh}\left (\frac {1}{c x}\right )\right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1+e^{-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {i}{2 x^2}\right )}{b}\)

Input:

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

Output:

((-I)*((-1/2*I)/x^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[1/(c*x)])*Log[1 + E^(( 
2*a)/b - I*Pi - (2*(a + b*ArcSinh[1/(c*x)]))/b)]) + (b^2*PolyLog[2, -a - b 
*ArcSinh[1/(c*x)]])/4)))/b
 

Defintions of rubi rules used

rule 25

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

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6836

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + 
b*ArcSinh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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