3.158 \(\int \cosh (a+b x) \coth (c+b x) \, dx\)

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

[Out]

-arctanh(cosh(b*x+c))*cosh(a-c)/b+cosh(b*x+a)/b

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 = {5621, 2638, 3770} \[ \frac {\cosh (a+b x)}{b}-\frac {\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]*Coth[c + b*x],x]

[Out]

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

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]

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]

Rubi steps

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

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Mathematica [C]  time = 0.06, size = 93, normalized size = 3.21 \[ -\frac {2 i \cosh (a-c) \tan ^{-1}\left (\frac {(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac {b x}{2}\right )+\cosh (c) \cosh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \sinh (c) \cosh \left (\frac {b x}{2}\right )}\right )}{b}+\frac {\sinh (a) \sinh (b x)}{b}+\frac {\cosh (a) \cosh (b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.44, size = 439, normalized size = 15.14 \[ \frac {\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} + {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) + 1\right ) - {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) - 1\right ) + 2 \, {\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right ) + 1}{2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(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*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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giac [B]  time = 0.12, size = 85, normalized size = 2.93 \[ -\frac {{\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) - {\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) - e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.22, size = 155, normalized size = 5.34 \[ \frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.33, size = 90, normalized size = 3.10 \[ -\frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {e^{\left (b x + a\right )}}{2 \, b} + \frac {e^{\left (-b x - a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*coth(b*x+c),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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mupad [B]  time = 0.15, size = 139, normalized size = 4.79 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}+{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left (2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left (2\,{\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{4\,a-4\,c}+1\right )}}{\sqrt {-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*coth(c + b*x),x)

[Out]

exp(a + b*x)/(2*b) + exp(- a - b*x)/(2*b) - (atan((exp(-a)*exp(2*c)*exp(b*x)*((-b^2)^(1/2) + exp(2*a)*exp(-2*c
)*(-b^2)^(1/2)))/(b*(exp(-2*a)*exp(2*c)*(2*exp(2*a)*exp(-2*c) + exp(4*a)*exp(-4*c) + 1))^(1/2)))*(exp(2*c - 2*
a)*(2*exp(2*a - 2*c) + exp(4*a - 4*c) + 1))^(1/2))/(-b^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\left (a + b x \right )} \coth {\left (b x + c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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