Optimal. Leaf size=37 \[ -\frac{\coth (a-c) \log (\cosh (a+b x))}{b}+\frac{\coth (a-c) \log (\cosh (b x+c))}{b}+x \]
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Rubi [A] time = 0.0695035, antiderivative size = 37, normalized size of antiderivative = 1., 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 = {5646, 5644, 3475} \[ -\frac{\coth (a-c) \log (\cosh (a+b x))}{b}+\frac{\coth (a-c) \log (\cosh (b x+c))}{b}+x \]
Antiderivative was successfully verified.
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Rule 5646
Rule 5644
Rule 3475
Rubi steps
\begin{align*} \int \tanh (a+b x) \tanh (c+b x) \, dx &=x-\cosh (a-c) \int \text{sech}(a+b x) \text{sech}(c+b x) \, dx\\ &=x-\coth (a-c) \int \tanh (a+b x) \, dx+\coth (a-c) \int \tanh (c+b x) \, dx\\ &=x-\frac{\coth (a-c) \log (\cosh (a+b x))}{b}+\frac{\coth (a-c) \log (\cosh (c+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.472242, size = 29, normalized size = 0.78 \[ \frac{\coth (a-c) (\log (\cosh (b x+c))-\log (\cosh (a+b x)))}{b}+x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 151, normalized size = 4.1 \begin{align*} x-{\frac{\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ){{\rm e}^{2\,a}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }}-{\frac{\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ){{\rm e}^{2\,c}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{2\,a}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{2\,c}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.71078, size = 112, normalized size = 3.03 \begin{align*} x + \frac{a}{b} - \frac{{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \log \left (e^{\left (-2 \, b x - 2 \, 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 (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98148, size = 701, normalized size = 18.95 \begin{align*} \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 (b x + c\right ) \cosh \left (-a + c\right ) - \sinh \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 \, \cosh \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh{\left (a + b x \right )} \tanh{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.47571, size = 149, normalized size = 4.03 \begin{align*} \frac{b x - \frac{{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \log \left (\frac{{\left | -{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |} + 2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )} \right |}}{{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |} + 2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}}\right )}{{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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