3.140 \(\int \text {sech}(c-b x) \text {sech}(a+b x) \, dx\)

Optimal. Leaf size=33 \[ \frac {\text {csch}(a+c) \log (\cosh (a+b x))}{b}-\frac {\text {csch}(a+c) \log (\cosh (c-b x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5644, 3475} \[ \frac {\text {csch}(a+c) \log (\cosh (a+b x))}{b}-\frac {\text {csch}(a+c) \log (\cosh (c-b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c - b*x]*Sech[a + b*x],x]

[Out]

-((Csch[a + c]*Log[Cosh[c - b*x]])/b) + (Csch[a + c]*Log[Cosh[a + 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 5644

Int[Sech[(a_.) + (b_.)*(x_)]*Sech[(c_) + (d_.)*(x_)], x_Symbol] :> -Dist[Csch[(b*c - a*d)/d], Int[Tanh[a + b*x
], x], x] + Dist[Csch[(b*c - a*d)/b], Int[Tanh[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 {sech}(c-b x) \text {sech}(a+b x) \, dx &=\text {csch}(a+c) \int \tanh (c-b x) \, dx+\text {csch}(a+c) \int \tanh (a+b x) \, dx\\ &=-\frac {\text {csch}(a+c) \log (\cosh (c-b x))}{b}+\frac {\text {csch}(a+c) \log (\cosh (a+b x))}{b}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 27, normalized size = 0.82 \[ -\frac {\text {csch}(a+c) (\log (\cosh (c-b x))-\log (\cosh (a+b x)))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c - b*x]*Sech[a + b*x],x]

[Out]

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

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fricas [B]  time = 0.42, size = 156, normalized size = 4.73 \[ \frac {2 \, {\left ({\left (\cosh \left (a + c\right ) - \sinh \left (a + c\right )\right )} \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) \cosh \left (a + c\right ) - \sinh \left (b x + a\right ) \sinh \left (a + c\right )\right )}}{\cosh \left (b x + a\right ) \cosh \left (a + c\right ) - {\left (\cosh \left (a + c\right ) + \sinh \left (a + c\right )\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (a + c\right )}\right ) - {\left (\cosh \left (a + c\right ) - \sinh \left (a + c\right )\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )\right )}}{b \cosh \left (a + c\right )^{2} - 2 \, b \cosh \left (a + c\right ) \sinh \left (a + c\right ) + b \sinh \left (a + c\right )^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((cosh(a + c) - sinh(a + c))*log(2*(cosh(b*x + a)*cosh(a + c) - sinh(b*x + a)*sinh(a + c))/(cosh(b*x + a)*co
sh(a + c) - (cosh(a + c) + sinh(a + c))*sinh(b*x + a) + cosh(b*x + a)*sinh(a + c))) - (cosh(a + c) - sinh(a +
c))*log(2*cosh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))))/(b*cosh(a + c)^2 - 2*b*cosh(a + c)*sinh(a + c) + b*s
inh(a + c)^2 - b)

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giac [B]  time = 0.12, size = 70, normalized size = 2.12 \[ -\frac {2 \, {\left (\frac {e^{\left (a + c\right )} \log \left (e^{\left (2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}{e^{\left (2 \, a + 2 \, c\right )} - 1} + \frac {e^{\left (3 \, a + c\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{e^{\left (2 \, a\right )} - e^{\left (4 \, a + 2 \, c\right )}}\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.16, size = 75, normalized size = 2.27 \[ \frac {2 \ln \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{a +c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}-\frac {2 \ln \left ({\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{a +c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x-c)*sech(b*x+a),x)

[Out]

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

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maxima [A]  time = 0.40, size = 67, normalized size = 2.03 \[ \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} - \frac {2 \, e^{\left (a + c\right )} \log \left (e^{\left (-2 \, b x + 2 \, c\right )} + 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.37, size = 268, normalized size = 8.12 \[ \frac {4\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\left (\frac {2\,{\mathrm {e}}^a\,{\mathrm {e}}^c}{b\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}\right )}^{3/2}}+\frac {2\,{\mathrm {e}}^{-3\,a}\,{\mathrm {e}}^{-3\,c}\,\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}+1\right )\,\left (b\,\sqrt {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}}+b\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}\right )}^{3/2}\right )}{\sqrt {-b^2\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-1\right )}^2}\,\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-b^2-b^2\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,c}}}\right )\,\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-b^2-b^2\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,c}}}{4}+\frac {b\,{\mathrm {e}}^{-3\,a}\,{\mathrm {e}}^{-3\,c}\,\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}+1\right )\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}\right )}^{3/2}}{\sqrt {-b^2\,{\left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-1\right )}^2}}\right )\,\sqrt {{\mathrm {e}}^{2\,a+2\,c}}}{\sqrt {2\,b^2\,{\mathrm {e}}^{2\,a+2\,c}-b^2\,{\mathrm {e}}^{4\,a+4\,c}-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*atan((exp(2*a)*exp(2*b*x)*((2*exp(a)*exp(c))/(b*(exp(2*a)*exp(2*c))^(3/2)) + (2*exp(-3*a)*exp(-3*c)*(exp(2*
a)*exp(2*c) + 1)*(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*e
xp(4*a)*exp(4*c))^(1/2))/4 + (b*exp(-3*a)*exp(-3*c)*(exp(2*a)*exp(2*c) + 1)*(exp(2*a)*exp(2*c))^(3/2))/(-b^2*(
exp(2*a)*exp(2*c) - 1)^2)^(1/2))*exp(2*a + 2*c)^(1/2))/(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 \[ \int \operatorname {sech}{\left (a + b x \right )} \operatorname {sech}{\left (b x - c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x-c)*sech(b*x+a),x)

[Out]

Integral(sech(a + b*x)*sech(b*x - c), x)

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