Optimal. Leaf size=36 \[ -\frac {\coth (a+c) \log (\sinh (c-b x))}{b}+\frac {\coth (a+c) \log (\sinh (a+b x))}{b}-x \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5647, 5645, 3475} \[ -\frac {\coth (a+c) \log (\sinh (c-b x))}{b}+\frac {\coth (a+c) \log (\sinh (a+b x))}{b}-x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 5645
Rule 5647
Rubi steps
\begin {align*} \int \coth (c-b x) \coth (a+b x) \, dx &=-x+\cosh (a+c) \int \text {csch}(c-b x) \text {csch}(a+b x) \, dx\\ &=-x+\coth (a+c) \int \coth (c-b x) \, dx+\coth (a+c) \int \coth (a+b x) \, dx\\ &=-x-\frac {\coth (a+c) \log (\sinh (c-b x))}{b}+\frac {\coth (a+c) \log (\sinh (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 32, normalized size = 0.89 \[ \frac {\coth (a+c) (\log (-\sinh (a+b x))-\log (\sinh (c-b x)))}{b}-x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 216, normalized size = 6.00 \[ -\frac {b x \cosh \left (a + c\right )^{2} - 2 \, b x \cosh \left (a + c\right ) \sinh \left (a + c\right ) + b x \sinh \left (a + c\right )^{2} - b x - {\left (\cosh \left (a + c\right )^{2} - 2 \, \cosh \left (a + c\right ) \sinh \left (a + c\right ) + \sinh \left (a + c\right )^{2} + 1\right )} \log \left (\frac {2 \, {\left (\cosh \left (a + c\right ) \sinh \left (b x + a\right ) - \cosh \left (b x + a\right ) \sinh \left (a + c\right )\right )}}{\cosh \left (b x + a\right ) \cosh \left (a + c\right ) - {\left (\cosh \left (a + c\right ) + \sinh \left (a + c\right )\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (a + c\right )}\right ) + {\left (\cosh \left (a + c\right )^{2} - 2 \, \cosh \left (a + c\right ) \sinh \left (a + c\right ) + \sinh \left (a + c\right )^{2} + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b \cosh \left (a + c\right )^{2} - 2 \, b \cosh \left (a + c\right ) \sinh \left (a + c\right ) + b \sinh \left (a + c\right )^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 90, normalized size = 2.50 \[ -\frac {b x + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left ({\left | e^{\left (2 \, b x\right )} - e^{\left (2 \, c\right )} \right |}\right )}{e^{\left (2 \, a + 2 \, c\right )} - 1} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (4 \, a + 2 \, c\right )}\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{e^{\left (2 \, a\right )} - e^{\left (4 \, a + 2 \, c\right )}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 153, normalized size = 4.25 \[ -x -\frac {\ln \left (-{\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{2 a +2 c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}-\frac {\ln \left (-{\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right )}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{2 a +2 c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 a +2 c}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 160, normalized size = 4.44 \[ -x - \frac {a}{b} + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x - a\right )} + 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x - a\right )} - 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} - \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x + c\right )} + 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} - \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x + c\right )} - 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 121, normalized size = 3.36 \[ \frac {\mathrm {coth}\left (a+c\right )\,\ln \left (4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}+4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,c}-4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,b\,x}-4\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{2\,b\,x}\right )}{b}-\frac {\mathrm {coth}\left (a+c\right )\,\ln \left (4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,c}+4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,b\,x}\right )}{b}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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