### 3.101 $$\int \cosh (a+b x) \coth (a+b x) \, dx$$

Optimal. Leaf size=23 $\frac{\cosh (a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}$

[Out]

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

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Rubi [A]  time = 0.0180681, antiderivative size = 23, 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 = {2592, 321, 206} $\frac{\cosh (a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

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

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 321

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

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

Mathematica [A]  time = 0.0257619, size = 26, normalized size = 1.13 $\frac{\cosh (a+b x)}{b}+\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

Cosh[a + b*x]/b + Log[Tanh[(a + b*x)/2]]/b

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Maple [A]  time = 0.016, size = 21, normalized size = 0.9 \begin{align*}{\frac{\cosh \left ( bx+a \right ) -2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*coth(b*x+a),x)

[Out]

1/b*(cosh(b*x+a)-2*arctanh(exp(b*x+a)))

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Maxima [B]  time = 1.20723, size = 80, normalized size = 3.48 \begin{align*} \frac{e^{\left (b x + a\right )}}{2 \, b} + \frac{e^{\left (-b x - a\right )}}{2 \, b} - \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.84889, size = 355, normalized size = 15.43 \begin{align*} \frac{\cosh \left (b x + a\right )^{2} - 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}{2 \,{\left (b \cosh \left (b x + a\right ) + b \sinh \left (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+a),x, algorithm="fricas")

[Out]

1/2*(cosh(b*x + a)^2 - 2*(cosh(b*x + a) + sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 2*(cosh(b*x
+ a) + sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2
+ 1)/(b*cosh(b*x + a) + b*sinh(b*x + a))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18726, size = 59, normalized size = 2.57 \begin{align*} \frac{e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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