Optimal. Leaf size=36 \[ -x-\frac {\coth (a+c) \log (\sinh (c-b x))}{b}+\frac {\coth (a+c) \log (\sinh (a+b x))}{b} \]
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Rubi [A]
time = 0.03, 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 = {5766, 5764,
3556} \begin {gather*} -\frac {\coth (a+c) \log (\sinh (c-b x))}{b}+\frac {\coth (a+c) \log (\sinh (a+b x))}{b}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 5764
Rule 5766
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 = 0.34, size = 32, normalized size = 0.89 \begin {gather*} -x+\frac {\coth (a+c) (-\log (\sinh (c-b x))+\log (-\sinh (a+b x)))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(152\) vs.
\(2(39)=78\).
time = 1.35, size = 153, normalized size = 4.25
method | result | size |
risch | \(-x -\frac {\ln \left (-{\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{2 a +2 c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}-\frac {\ln \left (-{\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right )}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{2 a +2 c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs.
\(2 (39) = 78\).
time = 0.28, size = 160, normalized size = 4.44 \begin {gather*} -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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs.
\(2 (39) = 78\).
time = 0.37, size = 216, normalized size = 6.00 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} - \int \coth {\left (a + b x \right )} \coth {\left (b x - c \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (39) = 78\).
time = 0.40, size = 90, normalized size = 2.50 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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