\(\int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx\) [234]

Optimal result

Integrand size = 20, antiderivative size = 48 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {e^{a c+b c x} \cot ^{-1}(\text {csch}(c (a+b x)))}{b c}-\frac {\log \left (1+e^{2 c (a+b x)}\right )}{b c} \]

[Out]

exp(b*c*x+a*c)*arccot(csch(c*(b*x+a)))/b/c-ln(1+exp(2*c*(b*x+a)))/b/c

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 48, 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, 5316, 2320, 12, 266} \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {e^{a c+b c x} \cot ^{-1}(\text {csch}(c (a+b x)))}{b c}-\frac {\log \left (e^{2 c (a+b x)}+1\right )}{b c} \]

[In]

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

[Out]

(E^(a*c + b*c*x)*ArcCot[Csch[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 5316

Int[((a_.) + ArcCot[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcCot[u], w, x] + Dist
[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[Functi
onOfLinear[v*(a + b*ArcCot[u]), x]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \cot ^{-1}(\text {csch}(x)) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {csch}(c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int e^x \text {sech}(x) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {csch}(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} \cot ^{-1}(\text {csch}(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} \cot ^{-1}(\text {csch}(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 = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {e^{c (a+b x)} \cot ^{-1}\left (\frac {2 e^{c (a+b x)}}{-1+e^{2 c (a+b x)}}\right )-\log \left (1+e^{2 c (a+b x)}\right )}{b c} \]

[In]

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

[Out]

(E^(c*(a + b*x))*ArcCot[(2*E^(c*(a + b*x)))/(-1 + E^(2*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 = 1.35 (sec) , antiderivative size = 903, normalized size of antiderivative = 18.81

method	result	size
risch	\(\text {Expression too large to display}\)	\(903\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.56 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \arctan \left (\sinh \left (b c x + a c\right )\right ) - \log \left (\frac {2 \, \cosh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \]

[In]

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

[Out]

((cosh(b*c*x + a*c) + sinh(b*c*x + a*c))*arctan(sinh(b*c*x + a*c)) - log(2*cosh(b*c*x + a*c)/(cosh(b*c*x + a*c
) - sinh(b*c*x + a*c))))/(b*c)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {\operatorname {arccot}\left (\operatorname {csch}\left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} - \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{b c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.35 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {{\left (\arctan \left (\frac {1}{2} \, e^{\left (b c x + a c\right )} - \frac {1}{2} \, e^{\left (-b c x - a c\right )}\right ) e^{\left (b c x\right )} - e^{\left (-a c\right )} \log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )\right )} e^{\left (a c\right )}}{b c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.42 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {csch}(a c+b c x)) \, dx=\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}\,\mathrm {acot}\left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{b\,c}-\frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}+1\right )}{b\,c} \]

[In]

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

[Out]

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