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

3.1.25.1 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(b*x+a)+( 
1+1/c^2/exp(b*x+a)^2)^(1/2))^2)/b-1/2*polylog(2,(1/c/exp(b*x+a)+(1+1/c^2/e 
xp(b*x+a)^2)^(1/2))^2)/b
 

3.1.25.2 Mathematica [B] (verified)

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

Time = 0.71 (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)))])
 

3.1.25.3 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
 

3.1.25.3.1 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]
 

3.1.25.4 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)
 

3.1.25.5 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)
 

3.1.25.6 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)
 

3.1.25.7 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
 

3.1.25.8 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)
 

3.1.25.9 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)