### 3.146 $$\int \coth (c+b x) \sinh (a+b x) \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0204265, 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 = {5622, 2637, 3770} $\frac{\sinh (a+b x)}{b}-\frac{\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[c + b*x]*Sinh[a + b*x],x]

[Out]

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

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]

Rubi steps

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

Mathematica [C]  time = 0.0530698, size = 93, normalized size = 3.21 $-\frac{2 i \sinh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac{b x}{2}\right )+\cosh (c) \cosh \left (\frac{b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac{b x}{2}\right )-i \sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}+\frac{\sinh (a) \cosh (b x)}{b}+\frac{\cosh (a) \sinh (b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[c + b*x]*Sinh[a + b*x],x]

[Out]

(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

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Maple [B]  time = 0.041, size = 155, normalized size = 5.3 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{2\,b}}-{\frac{{{\rm e}^{-bx-a}}}{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}}+{\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(coth(b*x+c)*sinh(b*x+a),x)

[Out]

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

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Maxima [B]  time = 1.25286, size = 127, normalized size = 4.38 \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{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+c)*sinh(b*x+a),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*e^(-b*x - a)/b

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Fricas [B]  time = 2.33157, size = 1233, normalized size = 42.52 \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} +{\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 )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) + 1\right ) -{\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 )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) - 1\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(coth(b*x+c)*sinh(b*x+a),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*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) - (cosh(-a + c)^2 -
2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c))*log(cosh(b*x + c) + sinh(b*x + c) + 1) - (2*c
osh(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) - (
cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c))*log(cosh(b*x + c) + sinh(b*x
+ c) - 1) + 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 \sinh{\left (a + b x \right )} \coth{\left (b x + c \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+c)*sinh(b*x+a),x)

[Out]

Integral(sinh(a + b*x)*coth(b*x + c), x)

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Giac [B]  time = 1.16616, size = 117, normalized size = 4.03 \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 ) - 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(coth(b*x+c)*sinh(b*x+a),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)) - e^(b*x + a) + e^(-b*x - a))/b