### 3.155 $$\int \cosh (a+b x) \tanh (c+b x) \, dx$$

Optimal. Leaf size=29 $\frac{\cosh (a+b x)}{b}-\frac{\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}$

[Out]

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

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Rubi [A]  time = 0.0193314, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.231, Rules used = {5623, 2638, 3770} $\frac{\cosh (a+b x)}{b}-\frac{\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Tanh[c + b*x],x]

[Out]

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

Rule 5623

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[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]

Rubi steps

\begin{align*} \int \cosh (a+b x) \tanh (c+b x) \, dx &=-(\sinh (a-c) \int \text{sech}(c+b x) \, dx)+\int \sinh (a+b x) \, dx\\ &=\frac{\cosh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (c+b x)) \sinh (a-c)}{b}\\ \end{align*}

Mathematica [B]  time = 0.057125, size = 86, normalized size = 2.97 $-\frac{2 \sinh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \cosh \left (\frac{b x}{2}\right )+\cosh (c) \sinh \left (\frac{b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac{b x}{2}\right )-\sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}+\frac{\sinh (a) \sinh (b x)}{b}+\frac{\cosh (a) \cosh (b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[a + b*x]*Tanh[c + b*x],x]

[Out]

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

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Maple [C]  time = 0.08, size = 167, normalized size = 5.8 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{2\,b}}+{\frac{{{\rm e}^{-bx-a}}}{2\,b}}+{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{a}} \right ) ^{2}}{b}}-{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{c}} \right ) ^{2}}{b}}-{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{a}} \right ) ^{2}}{b}}+{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{c}} \right ) ^{2}}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*tanh(b*x+c),x)

[Out]

1/2*exp(b*x+a)/b+1/2*exp(-b*x-a)/b+1/2*I/b*ln(exp(b*x+a)-I*exp(a-c))*exp(-a-c)*exp(a)^2-1/2*I/b*ln(exp(b*x+a)-
I*exp(a-c))*exp(-a-c)*exp(c)^2-1/2*I/b*ln(exp(b*x+a)+I*exp(a-c))*exp(-a-c)*exp(a)^2+1/2*I/b*ln(exp(b*x+a)+I*ex
p(a-c))*exp(-a-c)*exp(c)^2

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Maxima [B]  time = 1.64584, size = 80, normalized size = 2.76 \begin{align*} \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} + \frac{e^{\left (b x + a\right )}}{2 \, b} + \frac{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)*tanh(b*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.15508, size = 914, normalized size = 31.52 \begin{align*} \frac{\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right )^{2} +{\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} + 2 \,{\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} -{\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) -{\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )\right )} \arctan \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right )\right ) + 2 \,{\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right ) + 1}{2 \,{\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right ) +{\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cosh(b*x + c)^2*cosh(-a + c)^2 - 2*cosh(b*x + c)^2*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)^2*sinh(-a +
c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2)*sinh(b*x + c)^2 + 2*(2*cosh(b*x + c)*co
sh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c) - (cosh(-a + c)^2
- 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c))*arctan(cosh(b*x + c) + sinh(b*x + c)) + 2*(
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) + 1)/(b*cosh(b*x + c)*cosh(-a + c) - b*cosh(b*x + c)*sinh(-a + c) + (b*cosh(-a + c) - b*sinh(-a + c))
*sinh(b*x + c))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*tanh(b*x+c),x)

[Out]

Integral(cosh(a + b*x)*tanh(b*x + c), x)

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Giac [A]  time = 1.18435, size = 72, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - e^{\left (b x + a\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)*tanh(b*x+c),x, algorithm="giac")

[Out]

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