3.141 \(\int \text {csch}(a+b x) \text {csch}(c+b x) \, dx\)

Optimal. Leaf size=36 \[ \frac {\text {csch}(a-c) \log (\sinh (b x+c))}{b}-\frac {\text {csch}(a-c) \log (\sinh (a+b x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5645, 3475} \[ \frac {\text {csch}(a-c) \log (\sinh (b x+c))}{b}-\frac {\text {csch}(a-c) \log (\sinh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]*Csch[c + b*x],x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.23, size = 28, normalized size = 0.78 \[ -\frac {\text {csch}(a-c) (\log (\sinh (a+b x))-\log (\sinh (b x+c)))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.53, size = 184, normalized size = 5.11 \[ -\frac {2 \, {\left ({\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac {2 \, {\left (\cosh \left (-a + c\right ) \sinh \left (b x + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - {\left (\cosh \left (-a + c\right ) + \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )}\right ) - {\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac {2 \, \sinh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )\right )}}{b \cosh \left (-a + c\right )^{2} - 2 \, b \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + b \sinh \left (-a + c\right )^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((cosh(-a + c) - sinh(-a + c))*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) -
sinh(-a + c))*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 = 0.12, size = 81, normalized size = 2.25 \[ -\frac {2 \, {\left (\frac {e^{\left (3 \, a + c\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{e^{\left (4 \, a\right )} - e^{\left (2 \, a + 2 \, c\right )}} - \frac {e^{\left (a + 3 \, c\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, c\right )} - 1 \right |}\right )}{e^{\left (2 \, a + 2 \, c\right )} - e^{\left (4 \, c\right )}}\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(e^(3*a + c)*log(abs(e^(2*b*x + 2*a) - 1))/(e^(4*a) - e^(2*a + 2*c)) - e^(a + 3*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 = 0.13, size = 79, normalized size = 2.19 \[ \frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{a +c}}{\left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right ) b}-\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{a +c}}{\left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.32, size = 133, normalized size = 3.69 \[ -\frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x - a\right )} + 1\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} - \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x - a\right )} - 1\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} + \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} + \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{b {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.77, size = 266, normalized size = 7.39 \[ -\frac {4\,\sqrt {{\mathrm {e}}^{2\,a-2\,c}}\,\mathrm {atan}\left (\frac {b\,\left ({\mathrm {e}}^{-a}\,{\mathrm {e}}^c+{\mathrm {e}}^{-3\,a}\,{\mathrm {e}}^{3\,c}\right )\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )}^{3/2}}{\sqrt {-b^2\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-1\right )}^2}}-\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\left (\frac {2\,{\mathrm {e}}^{-c}\,{\mathrm {e}}^a}{b\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )}^{3/2}}+\frac {2\,\left ({\mathrm {e}}^{-a}\,{\mathrm {e}}^c+{\mathrm {e}}^{-3\,a}\,{\mathrm {e}}^{3\,c}\right )\,\left (b\,\sqrt {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}}+b\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )}^{3/2}\right )}{\sqrt {-b^2\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-1\right )}^2}\,\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-b^2-b^2\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}}}\right )\,\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}-b^2-b^2\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}}}{4}\right )}{\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a-2\,c}-b^2\,{\mathrm {e}}^{4\,a-4\,c}-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(a + b*x)*sinh(c + b*x)),x)

[Out]

-(4*exp(2*a - 2*c)^(1/2)*atan((b*(exp(-a)*exp(c) + exp(-3*a)*exp(3*c))*(exp(2*a)*exp(-2*c))^(3/2))/(-b^2*(exp(
2*a)*exp(-2*c) - 1)^2)^(1/2) - (exp(2*a)*exp(2*b*x)*((2*exp(-c)*exp(a))/(b*(exp(2*a)*exp(-2*c))^(3/2)) + (2*(e
xp(-a)*exp(c) + exp(-3*a)*exp(3*c))*(b*(exp(2*a)*exp(-2*c))^(1/2) + b*(exp(2*a)*exp(-2*c))^(3/2)))/((-b^2*(exp
(2*a)*exp(-2*c) - 1)^2)^(1/2)*(2*b^2*exp(2*a)*exp(-2*c) - b^2 - b^2*exp(4*a)*exp(-4*c))^(1/2)))*(2*b^2*exp(2*a
)*exp(-2*c) - b^2 - b^2*exp(4*a)*exp(-4*c))^(1/2))/4))/(2*b^2*exp(2*a - 2*c) - b^2*exp(4*a - 4*c) - b^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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