\(\int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx\) [359]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\frac {e^{a c+b c x} \text {arctanh}(\text {sech}(c (a+b x)))}{b c}+\frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c} \] Output:

exp(b*c*x+a*c)*arctanh(sech(c*(b*x+a)))/b/c+ln(1-exp(2*c*(b*x+a)))/b/c
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.20 \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\frac {e^{c (a+b x)} \text {arctanh}\left (\frac {2 e^{c (a+b x)}}{1+e^{2 c (a+b x)}}\right )+\log \left (1-e^{2 c (a+b x)}\right )}{b c} \] Input:

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

Output:

(E^(c*(a + b*x))*ArcTanh[(2*E^(c*(a + b*x)))/(1 + E^(2*c*(a + b*x)))] + Lo 
g[1 - E^(2*c*(a + b*x))])/(b*c)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7281, 6829, 25, 2720, 27, 240}

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.

\(\displaystyle \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx\)

\(\Big \downarrow \) 7281

\(\displaystyle \frac {\int e^{a c+b x c} \text {arctanh}(\text {sech}(a c+b x c))d(a c+b x c)}{b c}\)

\(\Big \downarrow \) 6829

\(\displaystyle \frac {e^{a c+b c x} \text {arctanh}(\text {sech}(a c+b c x))-\int -e^{a c+b x c} \text {csch}(a c+b x c)d(a c+b x c)}{b c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int e^{a c+b x c} \text {csch}(a c+b x c)d(a c+b x c)+e^{a c+b c x} \text {arctanh}(\text {sech}(a c+b c x))}{b c}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int -\frac {2 e^{a c+b x c}}{1-e^{2 a c+2 b x c}}de^{a c+b x c}+e^{a c+b c x} \text {arctanh}(\text {sech}(a c+b c x))}{b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^{a c+b c x} \text {arctanh}(\text {sech}(a c+b c x))-2 \int \frac {e^{a c+b x c}}{1-e^{2 a c+2 b x c}}de^{a c+b x c}}{b c}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {e^{a c+b c x} \text {arctanh}(\text {sech}(a c+b c x))+\log \left (1-e^{2 a c+2 b c x}\right )}{b c}\)

Input:

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

Output:

(E^(a*c + b*c*x)*ArcTanh[Sech[a*c + b*c*x]] + Log[1 - E^(2*a*c + 2*b*c*x)] 
)/(b*c)
 

Defintions of rubi rules used

rule 25

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

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 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

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 6829

Int[((a_.) + ArcTanh[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]} 
, Simp[(a + b*ArcTanh[u])   w, x] - Simp[b   Int[SimplifyIntegrand[w*(D[u, 
x]/(1 - u^2)), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x 
] && InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; F 
reeQ[{c, d, m}, x]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTanh[u]), x]]
 

rule 7281

Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] 
 Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /;  !FalseQ[lst]]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.72 (sec) , antiderivative size = 872, normalized size of antiderivative = 17.80

method	result	size
risch	\(\text {Expression too large to display}\)	\(872\)

Input:

int(exp(c*(b*x+a))*arctanh(sech(b*c*x+a*c)),x,method=_RETURNVERBOSE)
 

Output:

-1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))+1))^2*csgn(I*(exp(c*(b*x+a))+1)^2)*ex 
p(c*(b*x+a))-1/4*I/b/c*Pi*csgn(I/(1+exp(2*c*(b*x+a))))*csgn(I*(exp(c*(b*x+ 
a))-1)^2/(1+exp(2*c*(b*x+a))))^2*exp(c*(b*x+a))-1/2*I/b/c*Pi*csgn(I*(exp(c 
*(b*x+a))-1))*csgn(I*(exp(c*(b*x+a))-1)^2)^2*exp(c*(b*x+a))+1/4*I/b/c*Pi*c 
sgn(I*(exp(c*(b*x+a))+1)^2)*csgn(I*(exp(c*(b*x+a))+1)^2/(1+exp(2*c*(b*x+a) 
)))^2*exp(c*(b*x+a))+1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))-1))^2*csgn(I*(exp 
(c*(b*x+a))-1)^2)*exp(c*(b*x+a))-1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))+1)^2) 
^3*exp(c*(b*x+a))+1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))-1)^2)*csgn(I/(1+exp( 
2*c*(b*x+a))))*csgn(I*(exp(c*(b*x+a))-1)^2/(1+exp(2*c*(b*x+a))))*exp(c*(b* 
x+a))+1/4*I/b/c*Pi*csgn(I/(1+exp(2*c*(b*x+a))))*csgn(I*(exp(c*(b*x+a))+1)^ 
2/(1+exp(2*c*(b*x+a))))^2*exp(c*(b*x+a))+1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a 
))-1)^2/(1+exp(2*c*(b*x+a))))^3*exp(c*(b*x+a))-1/4*I/b/c*Pi*csgn(I*(exp(c* 
(b*x+a))+1)^2/(1+exp(2*c*(b*x+a))))^3*exp(c*(b*x+a))-1/4*I/b/c*Pi*csgn(I*( 
exp(c*(b*x+a))-1)^2)*csgn(I*(exp(c*(b*x+a))-1)^2/(1+exp(2*c*(b*x+a))))^2*e 
xp(c*(b*x+a))+1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))-1)^2)^3*exp(c*(b*x+a))+1 
/2*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))+1))*csgn(I*(exp(c*(b*x+a))+1)^2)^2*exp( 
c*(b*x+a))-1/4*I/b/c*Pi*csgn(I*(exp(c*(b*x+a))+1)^2)*csgn(I/(1+exp(2*c*(b* 
x+a))))*csgn(I*(exp(c*(b*x+a))+1)^2/(1+exp(2*c*(b*x+a))))*exp(c*(b*x+a))-1 
/b/c*exp(c*(b*x+a))*ln(exp(c*(b*x+a))-1)+1/b/c*exp(c*(b*x+a))*ln(exp(c*(b* 
x+a))+1)-2*a/b+1/b/c*ln(-1+exp(2*c*(b*x+a)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.88 \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (\frac {\cosh \left (b c x + a c\right ) + 1}{\cosh \left (b c x + a c\right ) - 1}\right ) + 2 \, \log \left (\frac {2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{2 \, b c} \] Input:

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

Output:

1/2*((cosh(b*c*x + a*c) + sinh(b*c*x + a*c))*log((cosh(b*c*x + a*c) + 1)/( 
cosh(b*c*x + a*c) - 1)) + 2*log(2*sinh(b*c*x + a*c)/(cosh(b*c*x + a*c) - s 
inh(b*c*x + a*c))))/(b*c)
 

Sympy [F(-1)]

Timed out. \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\text {Timed out} \] Input:

integrate(exp(c*(b*x+a))*atanh(sech(b*c*x+a*c)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\frac {\operatorname {artanh}\left (\operatorname {sech}\left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \] Input:

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

Output:

arctanh(sech(b*c*x + a*c))*e^((b*x + a)*c)/(b*c) + log(e^(b*c*x + a*c) + 1 
)/(b*c) + log(e^(b*c*x + a*c) - 1)/(b*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (47) = 94\).

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.00 \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\frac {e^{\left ({\left (b x + a\right )} c\right )} \log \left (-\frac {\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}} + 1}{\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}} - 1}\right )}{2 \, b c} + \frac {\log \left ({\left | e^{\left (2 \, b c x + 2 \, a c\right )} - 1 \right |}\right )}{b c} \] Input:

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

Output:

1/2*e^((b*x + a)*c)*log(-(2/(e^(b*c*x + a*c) + e^(-b*c*x - a*c)) + 1)/(2/( 
e^(b*c*x + a*c) + e^(-b*c*x - a*c)) - 1))/(b*c) + log(abs(e^(2*b*c*x + 2*a 
*c) - 1))/(b*c)
 

Mupad [B] (verification not implemented)

Time = 4.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.43 \[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}-1\right )}{b\,c}-\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\ln \left (1-\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{2\,b\,c}+\frac {\ln \left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}+1\right )\,{\mathrm {e}}^{a\,c+b\,c\,x}}{2\,b\,c} \] Input:

int(atanh(1/cosh(a*c + b*c*x))*exp(c*(a + b*x)),x)
 

Output:

log(exp(2*b*c*x)*exp(2*a*c) - 1)/(b*c) - (exp(a*c + b*c*x)*log(1 - 1/((exp 
(b*c*x)*exp(a*c))/2 + (exp(-b*c*x)*exp(-a*c))/2)))/(2*b*c) + (log(1/((exp( 
b*c*x)*exp(a*c))/2 + (exp(-b*c*x)*exp(-a*c))/2) + 1)*exp(a*c + b*c*x))/(2* 
b*c)
 

Reduce [F]

\[ \int e^{c (a+b x)} \text {arctanh}(\text {sech}(a c+b c x)) \, dx=e^{a c} \left (\int e^{b c x} \mathit {atanh} \left (\mathrm {sech}\left (b c x +a c \right )\right )d x \right ) \] Input:

int(exp(c*(b*x+a))*atanh(sech(b*c*x+a*c)),x)
 

Output:

e**(a*c)*int(e**(b*c*x)*atanh(sech(a*c + b*c*x)),x)