\(\int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx\) [298]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=-\frac {e^{a c+b c x}}{b c}+\frac {e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c} \]

[Out]

-exp(b*c*x+a*c)/b/c+exp(b*c*x+a*c)*arccoth(coth(c*(b*x+a)))/b/c

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2225, 6411} \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=\frac {e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}-\frac {e^{a c+b c x}}{b c} \]

[In]

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

[Out]

-(E^(a*c + b*c*x)/(b*c)) + (E^(a*c + b*c*x)*ArcCoth[Coth[c*(a + b*x)]])/(b*c)

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 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}(\coth (x)) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int e^x \, dx,x,a c+b c x\right )}{b c} \\ & = -\frac {e^{a c+b c x}}{b c}+\frac {e^{a c+b c x} \coth ^{-1}(\coth (c (a+b x)))}{b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=\frac {e^{c (a+b x)} \left (-1+\coth ^{-1}\left (\frac {1+e^{2 c (a+b x)}}{-1+e^{2 c (a+b x)}}\right )\right )}{b c} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{c \left (b x +a \right )} \left (\operatorname {arccoth}\left (\frac {1}{\tanh \left (c \left (b x +a \right )\right )}\right )-1\right )}{b c}\)	\(29\)
default	\(\frac {{\mathrm e}^{b c x +a c} \left (b c x +a c \right )-{\mathrm e}^{b c x +a c}+{\mathrm e}^{b c x +a c} \left (\operatorname {arccoth}\left (\coth \left (b c x +a c \right )\right )-b c x -a c \right )}{b c}\)	\(68\)
risch	\(\frac {{\mathrm e}^{c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}\right )}{b c}-\frac {i \left (\pi \operatorname {csgn}\left (i {\mathrm e}^{c \left (b x +a \right )}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )-2 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{c \left (b x +a \right )}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right )^{3}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 c \left (b x +a \right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 c \left (b x +a \right )}}{{\mathrm e}^{2 c \left (b x +a \right )}-1}\right )^{3}-4 i\right ) {\mathrm e}^{c \left (b x +a \right )}}{4 b c}\)	\(302\)

[In]

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

[Out]

exp(c*(b*x+a))*(arccoth(1/tanh(c*(b*x+a)))-1)/b/c

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.56 \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=\frac {{\left (b c x + a c - 1\right )} e^{\left (b c x + a c\right )}}{b c} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=\frac {a e^{\left (b c x + a c\right )}}{b} + \frac {{\left (b c x e^{\left (a c\right )} - e^{\left (a c\right )}\right )} e^{\left (b c x\right )}}{b c} \]

[In]

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

[Out]

a*e^(b*c*x + a*c)/b + (b*c*x*e^(a*c) - e^(a*c))*e^(b*c*x)/(b*c)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=\frac {{\left (b^{2} c^{2} x + a b c^{2} - b c\right )} e^{\left (b c x + a c\right )}}{b^{2} c^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.62 \[ \int e^{c (a+b x)} \coth ^{-1}(\coth (a c+b c x)) \, dx=\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\left (\mathrm {acoth}\left (\mathrm {coth}\left (a\,c+b\,c\,x\right )\right )-1\right )}{b\,c} \]

[In]

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

[Out]

(exp(a*c + b*c*x)*(acoth(coth(a*c + b*c*x)) - 1))/(b*c)