### 3.159 $$\int \cosh (a+b x) \coth ^2(c+b x) \, dx$$

Optimal. Leaf size=46 $-\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}+\frac{\sinh (a+b x)}{b}$

[Out]

-((Cosh[a - c]*Csch[c + b*x])/b) - (ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b + Sinh[a + b*x]/b

________________________________________________________________________________________

Rubi [A]  time = 0.0426559, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {5621, 5622, 2637, 3770, 2606, 8} $-\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}+\frac{\sinh (a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Coth[c + b*x]^2,x]

[Out]

-((Cosh[a - c]*Csch[c + b*x])/b) - (ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b + Sinh[a + b*x]/b

Rule 5621

Int[Cosh[v_]*Coth[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Coth[w]^(n - 1), x] + Dist[Cosh[v - w], Int[Csch[w]*Coth
[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]

Rule 5622

Int[Coth[w_]^(n_.)*Sinh[v_], x_Symbol] :> Int[Cosh[v]*Coth[w]^(n - 1), x] + Dist[Sinh[v - w], Int[Csch[w]*Coth
[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cosh (a+b x) \coth ^2(c+b x) \, dx &=\cosh (a-c) \int \coth (c+b x) \text{csch}(c+b x) \, dx+\int \coth (c+b x) \sinh (a+b x) \, dx\\ &=-\frac{(i \cosh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(c+b x))}{b}+\sinh (a-c) \int \text{csch}(c+b x) \, dx+\int \cosh (a+b x) \, dx\\ &=-\frac{\cosh (a-c) \text{csch}(c+b x)}{b}-\frac{\tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{b}+\frac{\sinh (a+b x)}{b}\\ \end{align*}

Mathematica [C]  time = 0.0955164, size = 110, normalized size = 2.39 $-\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{2 i \sinh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac{b x}{2}\right )+\cosh (c) \cosh \left (\frac{b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac{b x}{2}\right )-i \sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}+\frac{\sinh (a) \cosh (b x)}{b}+\frac{\cosh (a) \sinh (b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[a + b*x]*Coth[c + b*x]^2,x]

[Out]

-((Cosh[a - c]*Csch[c + b*x])/b) + (Cosh[b*x]*Sinh[a])/b - ((2*I)*ArcTan[((Cosh[c] - Sinh[c])*(Cosh[c]*Cosh[(b
*x)/2] + Sinh[c]*Sinh[(b*x)/2]))/(I*Cosh[c]*Cosh[(b*x)/2] - I*Cosh[(b*x)/2]*Sinh[c])]*Sinh[a - c])/b + (Cosh[a
]*Sinh[b*x])/b

________________________________________________________________________________________

Maple [B]  time = 0.047, size = 195, normalized size = 4.2 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{2\,b}}-{\frac{{{\rm e}^{-bx-a}}}{2\,b}}+{\frac{{{\rm e}^{bx+a}} \left ({{\rm e}^{2\,a}}+{{\rm e}^{2\,c}} \right ) }{b \left ( -{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) }}-{\frac{\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}-{\frac{\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*coth(b*x+c)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.15535, size = 194, normalized size = 4.22 \begin{align*} -\frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} - \frac{{\left (3 \, e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (2 \, c\right )}}{2 \, b{\left (e^{\left (-b x - a + 2 \, c\right )} - e^{\left (-3 \, b x - a\right )}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b + 1/2*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) -
e^c)/b - 1/2*e^(-b*x - a)/b - 1/2*((3*e^(2*a) + 2*e^(2*c))*e^(-2*b*x - 2*a) - e^(2*c))/(b*(e^(-b*x - a + 2*c)
- e^(-3*b*x - a)))

________________________________________________________________________________________

Fricas [B]  time = 2.06523, size = 3363, normalized size = 73.11 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cosh(b*x + c)^4*cosh(-a + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2)*sinh(b*x
+ c)^4 + 4*(cosh(b*x + c)*cosh(-a + c)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)*sinh(-a
+ c)^2)*sinh(b*x + c)^3 - 3*(cosh(-a + c)^2 + 1)*cosh(b*x + c)^2 + 3*(2*cosh(b*x + c)^2*cosh(-a + c)^2 + (2*co
sh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(2*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a
+ c) - 1)*sinh(b*x + c)^2 + (cosh(b*x + c)^4 - 3*cosh(b*x + c)^2)*sinh(-a + c)^2 - ((cosh(-a + c)^2 - 1)*cosh
(b*x + c)^3 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^3 - 3*(2*cosh(
b*x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b
*x + c)^2 + (cosh(b*x + c)^3 - cosh(b*x + c))*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c) + (3*(cosh(-
a + c)^2 - 1)*cosh(b*x + c)^2 + (3*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(3*cosh(b*x + c)^2
*cosh(-a + c) - cosh(-a + c))*sinh(-a + c) + 1)*sinh(b*x + c) - 2*(cosh(b*x + c)^3*cosh(-a + c) - cosh(b*x + c
)*cosh(-a + c))*sinh(-a + c))*log(cosh(b*x + c) + sinh(b*x + c) + 1) + ((cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 +
(cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^3 - 3*(2*cosh(b*x + c)*cosh
(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^2 + (
cosh(b*x + c)^3 - cosh(b*x + c))*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c) + (3*(cosh(-a + c)^2 - 1)
*cosh(b*x + c)^2 + (3*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(3*cosh(b*x + c)^2*cosh(-a + c)
- cosh(-a + c))*sinh(-a + c) + 1)*sinh(b*x + c) - 2*(cosh(b*x + c)^3*cosh(-a + c) - cosh(b*x + c)*cosh(-a + c
))*sinh(-a + c))*log(cosh(b*x + c) + sinh(b*x + c) - 1) + 2*(2*cosh(b*x + c)^3*cosh(-a + c)^2 + (2*cosh(b*x +
c)^3 - 3*cosh(b*x + c))*sinh(-a + c)^2 - 3*(cosh(-a + c)^2 + 1)*cosh(b*x + c) - 2*(2*cosh(b*x + c)^3*cosh(-a +
c) - 3*cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*sinh(b*x + c) - 2*(cosh(b*x + c)^4*cosh(-a + c) - 3*cosh(b*x
+ c)^2*cosh(-a + c))*sinh(-a + c) + 1)/(b*cosh(b*x + c)^3*cosh(-a + c) + (b*cosh(-a + c) - b*sinh(-a + c))*si
nh(b*x + c)^3 - b*cosh(b*x + c)*cosh(-a + c) + 3*(b*cosh(b*x + c)*cosh(-a + c) - b*cosh(b*x + c)*sinh(-a + c))
*sinh(b*x + c)^2 + (3*b*cosh(b*x + c)^2*cosh(-a + c) - b*cosh(-a + c) - (3*b*cosh(b*x + c)^2 - b)*sinh(-a + c)
)*sinh(b*x + c) - (b*cosh(b*x + c)^3 - b*cosh(b*x + c))*sinh(-a + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x \right )} \coth ^{2}{\left (b x + c \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(b*x+c)**2,x)

[Out]

Integral(cosh(a + b*x)*coth(b*x + c)**2, x)

________________________________________________________________________________________

Giac [B]  time = 1.18202, size = 169, normalized size = 3.67 \begin{align*} -\frac{{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) -{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) + \frac{{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, c\right )} - 1\right )} e^{\left (-a\right )}}{e^{\left (3 \, b x + 2 \, c\right )} - e^{\left (b x\right )}} - e^{\left (b x + a\right )}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(e^(b*x + c) + 1) - (e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(abs
(e^(b*x + c) - 1)) + (2*e^(2*b*x + 2*a) + 3*e^(2*b*x + 2*c) - 1)*e^(-a)/(e^(3*b*x + 2*c) - e^(b*x)) - e^(b*x +
a))/b