\(\int \frac {(a+b \text {csch}^{-1}(c x))^2}{x} \, dx\) [19]

Optimal result

Integrand size = 14, antiderivative size = 81 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x} \, dx=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-b \left (a+b \text {csch}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right ) \] Output:

1/3*(a+b*arccsch(c*x))^3/b-(a+b*arccsch(c*x))^2*ln(1-(1/c/x+(1+1/c^2/x^2)^ 
(1/2))^2)-b*(a+b*arccsch(c*x))*polylog(2,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)+1/ 
2*b^2*polylog(3,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x} \, dx=a b \text {csch}^{-1}(c x)^2+\frac {1}{3} b^2 \text {csch}^{-1}(c x)^3-2 a b \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-b^2 \text {csch}^{-1}(c x)^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+a^2 \log (c x)-b \left (a+b \text {csch}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right ) \] Input:

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

Output:

a*b*ArcCsch[c*x]^2 + (b^2*ArcCsch[c*x]^3)/3 - 2*a*b*ArcCsch[c*x]*Log[1 - E 
^(2*ArcCsch[c*x])] - b^2*ArcCsch[c*x]^2*Log[1 - E^(2*ArcCsch[c*x])] + a^2* 
Log[c*x] - b*(a + b*ArcCsch[c*x])*PolyLog[2, E^(2*ArcCsch[c*x])] + (b^2*Po 
lyLog[3, E^(2*ArcCsch[c*x])])/2
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6840, 3042, 26, 4199, 25, 2620, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

\(\displaystyle i \left (-2 i \left (b \left (\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )de^{2 \text {csch}^{-1}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle i \left (-2 i \left (b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2\right )-\frac {i \left (a+b \text {csch}^{-1}(c x)\right )^3}{3 b}\right )\)

Input:

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

Output:

I*(((-1/3*I)*(a + b*ArcCsch[c*x])^3)/b - (2*I)*(-1/2*((a + b*ArcCsch[c*x]) 
^2*Log[1 - E^(2*ArcCsch[c*x])]) + b*(-1/2*((a + b*ArcCsch[c*x])*PolyLog[2, 
 E^(2*ArcCsch[c*x])]) + (b*PolyLog[3, E^(2*ArcCsch[c*x])])/4)))
 

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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[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])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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