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

Optimal. Leaf size=37 \[ -\frac {\coth (a-c) \log (\sinh (a+b x))}{b}+\frac {\coth (a-c) \log (\sinh (b x+c))}{b}+x \]

[Out]

x-coth(a-c)*ln(sinh(b*x+a))/b+coth(a-c)*ln(sinh(b*x+c))/b

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5647, 5645, 3475} \[ -\frac {\coth (a-c) \log (\sinh (a+b x))}{b}+\frac {\coth (a-c) \log (\sinh (b x+c))}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]*Coth[c + b*x],x]

[Out]

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

Rule 3475

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

Rule 5645

Int[Csch[(a_.) + (b_.)*(x_)]*Csch[(c_) + (d_.)*(x_)], x_Symbol] :> Dist[Csch[(b*c - a*d)/b], Int[Coth[a + b*x]
, x], x] - Dist[Csch[(b*c - a*d)/d], Int[Coth[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0]
&& NeQ[b*c - a*d, 0]

Rule 5647

Int[Coth[(a_.) + (b_.)*(x_)]*Coth[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[(b*x)/d, x] + Dist[Cosh[(b*c - a*d)/d]
, Int[Csch[a + b*x]*Csch[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.50, size = 29, normalized size = 0.78 \[ \frac {\coth (a-c) (\log (\sinh (b x+c))-\log (\sinh (a+b x)))}{b}+x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 259, normalized size = 7.00 \[ \frac {b x \cosh \left (-a + c\right )^{2} - 2 \, b x \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + b x \sinh \left (-a + c\right )^{2} - b x - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \log \left (\frac {2 \, {\left (\cosh \left (-a + c\right ) \sinh \left (b x + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - {\left (\cosh \left (-a + c\right ) + \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )}\right ) + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )}{b \cosh \left (-a + c\right )^{2} - 2 \, b \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + b \sinh \left (-a + c\right )^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 97, normalized size = 2.62 \[ \frac {b x - \frac {{\left (e^{\left (4 \, a\right )} + e^{\left (2 \, a + 2 \, c\right )}\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{e^{\left (4 \, a\right )} - e^{\left (2 \, a + 2 \, c\right )}} + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, c\right )} - 1 \right |}\right )}{e^{\left (2 \, a + 2 \, c\right )} - e^{\left (4 \, c\right )}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x - (e^(4*a) + e^(2*a + 2*c))*log(abs(e^(2*b*x + 2*a) - 1))/(e^(4*a) - e^(2*a + 2*c)) + (e^(2*a + 2*c) + e^
(4*c))*log(abs(e^(2*b*x + 2*c) - 1))/(e^(2*a + 2*c) - e^(4*c)))/b

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maple [B]  time = 0.27, size = 155, normalized size = 4.19 \[ x -\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{2 a}}{b \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{2 c}}{b \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{2 a}}{b \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{2 c}}{b \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.40, size = 157, normalized size = 4.24 \[ x + \frac {a}{b} - \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \log \left (e^{\left (-b x - a\right )} + 1\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} - \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \log \left (e^{\left (-b x - a\right )} - 1\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.49, size = 115, normalized size = 3.11 \[ x-\frac {\ln \left (4\,{\mathrm {e}}^{4\,a}-4\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{2\,b\,x}+4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,b\,x}\right )\,\mathrm {coth}\left (a-c\right )}{b}+\frac {\ln \left (4\,{\mathrm {e}}^{4\,a}+4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{2\,b\,x}-4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,b\,x}\right )\,\mathrm {coth}\left (a-c\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*x)*coth(c + b*x),x)

[Out]

x - (log(4*exp(4*a) - 4*exp(6*a)*exp(2*b*x) + 4*exp(2*a)*exp(2*c) - 4*exp(4*a)*exp(2*c)*exp(2*b*x))*coth(a - c
))/b + (log(4*exp(4*a) + 4*exp(2*a)*exp(2*c) - 4*exp(2*a)*exp(4*c)*exp(2*b*x) - 4*exp(4*a)*exp(2*c)*exp(2*b*x)
)*coth(a - c))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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