\(\int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx\) [296]

Optimal result

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

[Out]

exp(b*c*x+a*c)*arccoth(cosh(c*(b*x+a)))/b/c+ln(1-exp(2*c*(b*x+a)))/b/c

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2225, 6411, 2320, 12, 266} \[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=\frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}+\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 6411

Int[((a_.) + ArcCoth[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcCoth[u], w, x] - Di
st[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_.) /; FreeQ[{c, d, m}, x]] && FalseQ[Func
tionOfLinear[v*(a + b*ArcCoth[u]), x]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \coth ^{-1}(\cosh (x)) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int e^x \text {csch}(x) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\text {Subst}\left (\int \frac {2 x}{-1+x^2} \, dx,x,e^{a c+b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {2 \text {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,e^{a c+b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=\frac {e^{c (a+b x)} \coth ^{-1}\left (\frac {1}{2} e^{-c (a+b x)} \left (1+e^{2 c (a+b x)}\right )\right )+\log \left (1-e^{2 c (a+b x)}\right )}{b c} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.36 (sec) , antiderivative size = 824, normalized size of antiderivative = 16.82

method	result	size
risch	\(\text {Expression too large to display}\)	\(824\)

[In]

int(exp(c*(b*x+a))*arccoth(cosh(b*c*x+a*c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.88 \[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=\frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (\frac {\cosh \left (b c x + a c\right ) + 1}{\cosh \left (b c x + a c\right ) - 1}\right ) + 2 \, \log \left (\frac {2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{2 \, b c} \]

[In]

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

[Out]

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*si
nh(b*c*x + a*c)/(cosh(b*c*x + a*c) - sinh(b*c*x + a*c))))/(b*c)

Sympy [F]

\[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=e^{a c} \int e^{b c x} \operatorname {acoth}{\left (\cosh {\left (a c + b c x \right )} \right )}\, dx \]

[In]

integrate(exp(c*(b*x+a))*acoth(cosh(b*c*x+a*c)),x)

[Out]

exp(a*c)*Integral(exp(b*c*x)*acoth(cosh(a*c + b*c*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=\frac {\operatorname {arcoth}\left (\cosh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \]

[In]

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

[Out]

arccoth(cosh(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 = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.00 \[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=\frac {e^{\left ({\left (b x + a\right )} c\right )} \log \left (-\frac {\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}} + 1}{\frac {2}{e^{\left (b c x + a c\right )} + e^{\left (-b c x - a c\right )}} - 1}\right )}{2 \, b c} + \frac {\log \left ({\left | e^{\left (2 \, b c x + 2 \, a c\right )} - 1 \right |}\right )}{b c} \]

[In]

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

[Out]

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 = 5.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.43 \[ \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}-1\right )}{b\,c}-\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\ln \left (1-\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{2\,b\,c}+\frac {\ln \left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}+1\right )\,{\mathrm {e}}^{a\,c+b\,c\,x}}{2\,b\,c} \]

[In]

int(exp(c*(a + b*x))*acoth(cosh(a*c + b*c*x)),x)

[Out]

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)*e
xp(-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)