\(\int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx\) [360]

Optimal result

Integrand size = 20, antiderivative size = 107 \[ \int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx=\frac {e^{a c+b c x} \text {arctanh}(\text {csch}(c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}-e^{2 c (a+b x)}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}-e^{2 c (a+b x)}\right )}{2 b c} \] Output:

exp(b*c*x+a*c)*arctanh(csch(c*(b*x+a)))/b/c+1/2*(1-2^(1/2))*ln(3-2*2^(1/2) 
-exp(2*c*(b*x+a)))/b/c+1/2*(1+2^(1/2))*ln(3+2*2^(1/2)-exp(2*c*(b*x+a)))/b/ 
c
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.40 \[ \int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx=\frac {-2 \sqrt {2} \text {arctanh}\left (\frac {-1+e^{c (a+b x)}}{\sqrt {2}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {1+e^{c (a+b x)}}{\sqrt {2}}\right )+2 e^{c (a+b x)} \text {arctanh}\left (\frac {2 e^{c (a+b x)}}{-1+e^{2 c (a+b x)}}\right )+\log \left (1-2 e^{c (a+b x)}-e^{2 c (a+b x)}\right )+\log \left (1+2 e^{c (a+b x)}-e^{2 c (a+b x)}\right )}{2 b c} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.50 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {7281, 6829, 25, 2720, 27, 1576, 1141, 2009}

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.

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

\(\Big \downarrow \) 7281

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

\(\Big \downarrow \) 6829

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int \frac {2 e^{a c+b x c} \left (1+e^{2 a c+2 b x c}\right )}{1-6 e^{2 a c+2 b x c}+e^{4 a c+4 b x c}}de^{a c+b x c}+e^{a c+b c x} \text {arctanh}(\text {csch}(a c+b c x))}{b c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {e^{a c+b x c} \left (1+e^{2 a c+2 b x c}\right )}{1-6 e^{2 a c+2 b x c}+e^{4 a c+4 b x c}}de^{a c+b x c}+e^{a c+b c x} \text {arctanh}(\text {csch}(a c+b c x))}{b c}\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {\int \frac {1+e^{2 a c+2 b x c}}{1-5 e^{2 a c+2 b x c}}de^{2 a c+2 b x c}+e^{a c+b c x} \text {arctanh}(\text {csch}(a c+b c x))}{b c}\)

\(\Big \downarrow \) 1141

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(E^(a*c + b*c*x)*ArcTanh[Csch[a*c + b*c*x]] + ((1 - Sqrt[2])*Log[3 - 2*Sqr 
t[2] - a*c - b*c*x])/2 + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] - a*c - b*c*x])/ 
2)/(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 1141

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[ 
(d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 
1] ||  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 
0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 = 4.47 (sec) , antiderivative size = 842, normalized size of antiderivative = 7.87

method	result	size
risch	\(\text {Expression too large to display}\)	\(842\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (90) = 180\).

Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.18 \[ \int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx=\frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (\frac {\sinh \left (b c x + a c\right ) + 1}{\sinh \left (b c x + a c\right ) - 1}\right ) + \sqrt {2} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (b c x + a c\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (b c x + a c\right )^{2} - 2 \, \sqrt {2} - 3}{\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} - 3}\right ) + \log \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} - 3\right )}}{\cosh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2}}\right )}{2 \, b c} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

exp(a*c)*Integral(exp(b*c*x)*atanh(csch(a*c + b*c*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (90) = 180\).

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.72 \[ \int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx=\frac {\operatorname {artanh}\left (\operatorname {csch}\left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (b c x + a c\right )} + 1}{\sqrt {2} + e^{\left (b c x + a c\right )} - 1}\right )}{2 \, b c} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (b c x + a c\right )} - 1}{\sqrt {2} + e^{\left (b c x + a c\right )} + 1}\right )}{2 \, b c} + \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )} - 1\right )}{2 \, b c} + \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} - 2 \, e^{\left (b c x + a c\right )} - 1\right )}{2 \, b c} \] Input:

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

Output:

arctanh(csch(b*c*x + a*c))*e^((b*x + a)*c)/(b*c) + 1/2*sqrt(2)*log(-(sqrt( 
2) - e^(b*c*x + a*c) + 1)/(sqrt(2) + e^(b*c*x + a*c) - 1))/(b*c) - 1/2*sqr 
t(2)*log(-(sqrt(2) - e^(b*c*x + a*c) - 1)/(sqrt(2) + e^(b*c*x + a*c) + 1)) 
/(b*c) + 1/2*log(e^(2*b*c*x + 2*a*c) + 2*e^(b*c*x + a*c) - 1)/(b*c) + 1/2* 
log(e^(2*b*c*x + 2*a*c) - 2*e^(b*c*x + a*c) - 1)/(b*c)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.56 \[ \int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx=\frac {e^{\left ({\left (b x + a\right )} c\right )} \log \left (-\frac {\frac {2}{e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}} + 1}{\frac {2}{e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}} - 1}\right )}{2 \, b c} + \frac {\sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} - 6 \right |}}\right ) + \log \left ({\left | e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1 \right |}\right )}{2 \, b c} \] Input:

integrate(exp(c*(b*x+a))*arctanh(csch(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) + 1/2*(sqrt(2)*log(abs(-4* 
sqrt(2) + 2*e^(2*b*c*x + 2*a*c) - 6)/abs(4*sqrt(2) + 2*e^(2*b*c*x + 2*a*c) 
 - 6)) + log(abs(e^(4*b*c*x + 4*a*c) - 6*e^(2*b*c*x + 2*a*c) + 1)))/(b*c)
 

Mupad [B] (verification not implemented)

Time = 4.01 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.75 \[ \int e^{c (a+b x)} \text {arctanh}(\text {csch}(a c+b c x)) \, dx=\frac {\ln \left (6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-2\,\sqrt {2}-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}+1\right )}{2\,b\,c}-\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\ln \left (1-\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{2\,b\,c}-\frac {\ln \left (2\,\sqrt {2}-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}-1\right )}{2\,b\,c}+\frac {\ln \left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}+1\right )\,{\mathrm {e}}^{a\,c+b\,c\,x}}{2\,b\,c} \] Input:

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

Output:

(log(6*2^(1/2)*exp(2*c*(a + b*x)) - 2*2^(1/2) - 8*exp(2*c*(a + b*x)))*(2^( 
1/2) + 1))/(2*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(2*2^(1/2) - 8*exp(2*c*(a + b 
*x)) - 6*2^(1/2)*exp(2*c*(a + b*x)))*(2^(1/2) - 1))/(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]

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

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

Output:

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