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]
________________________________________________________________________________________
Rubi [A] time = 0.0234886, antiderivative size = 33, normalized size of antiderivative = 1., 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]
[Out]
Rule 5644
Rule 3475
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*}
Mathematica [A] time = 0.199106, 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]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.033, size = 75, normalized size = 2.3 \begin{align*} -2\,{\frac{\ln \left ({{\rm e}^{2\,a+2\,c}}+{{\rm e}^{2\,bx+2\,a}} \right ){{\rm e}^{a+c}}}{b \left ({{\rm e}^{2\,a+2\,c}}-1 \right ) }}+2\,{\frac{\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ){{\rm e}^{a+c}}}{b \left ({{\rm e}^{2\,a+2\,c}}-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.61928, size = 90, normalized size = 2.73 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.14801, size = 471, normalized size = 14.27 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}{\left (a + b x \right )} \operatorname{sech}{\left (b x - c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17126, size = 95, normalized size = 2.88 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]