\(\int \text {csch}^{-1}(c e^{a+b x}) \, dx\) [25]

Optimal result

Integrand size = 10, antiderivative size = 77 \[ \int \text {csch}^{-1}\left (c e^{a+b x}\right ) \, dx=\frac {\text {csch}^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {\text {csch}^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}\left (c e^{a+b x}\right )}\right )}{2 b} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(236\) vs. \(2(77)=154\).

Time = 0.68 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.06 \[ \int \text {csch}^{-1}\left (c e^{a+b x}\right ) \, dx=x \text {csch}^{-1}\left (c e^{a+b x}\right )+\frac {e^{-a-b x} \sqrt {1+c^2 e^{2 (a+b x)}} \left (\log ^2\left (-c^2 e^{2 (a+b x)}\right )+\text {arctanh}\left (\sqrt {1+c^2 e^{2 (a+b x)}}\right ) \left (-8 b x+4 \log \left (-c^2 e^{2 (a+b x)}\right )\right )-4 \log \left (-c^2 e^{2 (a+b x)}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1+c^2 e^{2 (a+b x)}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1+c^2 e^{2 (a+b x)}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {1+c^2 e^{2 (a+b x)}}\right )\right )\right )}{8 b c \sqrt {1+\frac {e^{-2 (a+b x)}}{c^2}}} \] Input:

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

Output:

x*ArcCsch[c*E^(a + b*x)] + (E^(-a - b*x)*Sqrt[1 + c^2*E^(2*(a + b*x))]*(Lo 
g[-(c^2*E^(2*(a + b*x)))]^2 + ArcTanh[Sqrt[1 + c^2*E^(2*(a + b*x))]]*(-8*b 
*x + 4*Log[-(c^2*E^(2*(a + b*x)))]) - 4*Log[-(c^2*E^(2*(a + b*x)))]*Log[(1 
 + Sqrt[1 + c^2*E^(2*(a + b*x))])/2] + 2*Log[(1 + Sqrt[1 + c^2*E^(2*(a + b 
*x))])/2]^2 - 4*PolyLog[2, (1 - Sqrt[1 + c^2*E^(2*(a + b*x))])/2]))/(8*b*c 
*Sqrt[1 + 1/(c^2*E^(2*(a + b*x)))])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2720, 6836, 6190, 3042, 26, 4199, 25, 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.

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6836

\(\displaystyle -\frac {\int e^{-a-b x} \text {arcsinh}\left (\frac {e^{-a-b x}}{c}\right )de^{-a-b x}}{b}\)

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

\[\int \operatorname {arccsch}\left ({\mathrm e}^{b x +a} c \right )d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \text {csch}^{-1}\left (c e^{a+b x}\right ) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(asinh(exp(- a - b*x)/c),x)
                                                                                    
                                                                                    
 

Output:

int(asinh(exp(- a - b*x)/c), x)
 

Reduce [F]

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

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

Output:

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