3.213 \(\int e^{c (a+b x)} \sqrt{\coth ^2(a c+b c x)} \, dx\)

Optimal. Leaf size=83 \[ \frac{e^{c (a+b x)} \tanh (a c+b c x) \sqrt{\coth ^2(a c+b c x)}}{b c}-\frac{2 \tanh ^{-1}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x) \sqrt{\coth ^2(a c+b c x)}}{b c} \]

[Out]

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

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Rubi [A]  time = 0.143621, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6720, 2282, 388, 206} \[ \frac{e^{c (a+b x)} \tanh (a c+b c x) \sqrt{\coth ^2(a c+b c x)}}{b c}-\frac{2 \tanh ^{-1}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x) \sqrt{\coth ^2(a c+b c x)}}{b c} \]

Antiderivative was successfully verified.

[In]

Int[E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2],x]

[Out]

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

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 2282

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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int e^{c (a+b x)} \sqrt{\coth ^2(a c+b c x)} \, dx &=\left (\sqrt{\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \int e^{c (a+b x)} \coth (a c+b c x) \, dx\\ &=\frac{\left (\sqrt{\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{-1-x^2}{1-x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{e^{c (a+b x)} \sqrt{\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac{\left (2 \sqrt{\coth ^2(a c+b c x)} \tanh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{e^{c (a+b x)} \sqrt{\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac{2 \tanh ^{-1}\left (e^{c (a+b x)}\right ) \sqrt{\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}\\ \end{align*}

Mathematica [A]  time = 0.0577475, size = 51, normalized size = 0.61 \[ \frac{\left (e^{c (a+b x)}-2 \tanh ^{-1}\left (e^{c (a+b x)}\right )\right ) \tanh (c (a+b x)) \sqrt{\coth ^2(c (a+b x))}}{b c} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2],x]

[Out]

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

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Maple [B]  time = 0.238, size = 213, normalized size = 2.6 \begin{align*}{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ){{\rm e}^{c \left ( bx+a \right ) }}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) cb}\sqrt{{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}}+{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}-1 \right ) }{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) cb}\sqrt{{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}}-{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}+1 \right ) }{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) cb}\sqrt{{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))*(coth(b*c*x+a*c)^2)^(1/2),x)

[Out]

1/(1+exp(2*c*(b*x+a)))*(exp(2*c*(b*x+a))-1)*((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2)^(1/2)/c/b*exp(c*(b
*x+a))+1/(1+exp(2*c*(b*x+a)))*(exp(2*c*(b*x+a))-1)*((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2)^(1/2)/c/b*l
n(exp(c*(b*x+a))-1)-1/(1+exp(2*c*(b*x+a)))*(exp(2*c*(b*x+a))-1)*((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2
)^(1/2)/c/b*ln(exp(c*(b*x+a))+1)

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Maxima [A]  time = 1.70541, size = 76, normalized size = 0.92 \begin{align*} \frac{e^{\left (b c x + a 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(coth(b*c*x+a*c)^2)^(1/2),x, algorithm="maxima")

[Out]

e^(b*c*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)

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Fricas [A]  time = 2.37335, size = 196, normalized size = 2.36 \begin{align*} \frac{\cosh \left (b c x + a c\right ) - \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) + 1\right ) + \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) - 1\right ) + \sinh \left (b c x + a c\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(coth(b*c*x+a*c)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(coth(b*c*x+a*c)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.1916, size = 127, normalized size = 1.53 \begin{align*} \frac{\frac{e^{\left (b c x + a c\right )}}{\mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} - \frac{\log \left (e^{\left (b c x + a c\right )} + 1\right )}{\mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} + \frac{\log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right )}{\mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(coth(b*c*x+a*c)^2)^(1/2),x, algorithm="giac")

[Out]

(e^(b*c*x + a*c)/sgn(e^(2*b*c*x + 2*a*c) - 1) - log(e^(b*c*x + a*c) + 1)/sgn(e^(2*b*c*x + 2*a*c) - 1) + log(ab
s(e^(b*c*x + a*c) - 1))/sgn(e^(2*b*c*x + 2*a*c) - 1))/(b*c)