Optimal. Leaf size=42 \[ -\frac {\coth ^2(a+b x)}{2 b}-\frac {\coth ^4(a+b x)}{4 b}+\frac {\log (\sinh (a+b x))}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556}
\begin {gather*} -\frac {\coth ^4(a+b x)}{4 b}-\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rubi steps
\begin {align*} \int \coth ^5(a+b x) \, dx &=-\frac {\coth ^4(a+b x)}{4 b}+\int \coth ^3(a+b x) \, dx\\ &=-\frac {\coth ^2(a+b x)}{2 b}-\frac {\coth ^4(a+b x)}{4 b}+\int \coth (a+b x) \, dx\\ &=-\frac {\coth ^2(a+b x)}{2 b}-\frac {\coth ^4(a+b x)}{4 b}+\frac {\log (\sinh (a+b x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 44, normalized size = 1.05 \begin {gather*} -\frac {2 \coth ^2(a+b x)+\coth ^4(a+b x)-4 \log (\cosh (a+b x))-4 \log (\tanh (a+b x))}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 48, normalized size = 1.14
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\coth ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\coth ^{2}\left (b x +a \right )\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(48\) |
default | \(\frac {-\frac {\left (\coth ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\coth ^{2}\left (b x +a \right )\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(48\) |
risch | \(-x -\frac {2 a}{b}-\frac {4 \,{\mathrm e}^{2 b x +2 a} \left ({\mathrm e}^{4 b x +4 a}-{\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(76\) |
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. 122 vs.
\(2 (38) = 76\).
time = 0.26, size = 122, normalized size = 2.90 \begin {gather*} x + \frac {a}{b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\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. 978 vs.
\(2 (38) = 76\).
time = 0.42, size = 978, normalized size = 23.29 \begin {gather*} -\frac {b x \cosh \left (b x + a\right )^{8} + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b x \sinh \left (b x + a\right )^{8} - 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{6} + 4 \, {\left (7 \, b x \cosh \left (b x + a\right )^{2} - b x + 1\right )} \sinh \left (b x + a\right )^{6} + 8 \, {\left (7 \, b x \cosh \left (b x + a\right )^{3} - 3 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b x \cosh \left (b x + a\right )^{4} - 30 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 3 \, b x - 2\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b x \cosh \left (b x + a\right )^{5} - 10 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} + {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 4 \, {\left (7 \, b x \cosh \left (b x + a\right )^{6} - 15 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 3 \, {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )^{2} - b x + 1\right )} \sinh \left (b x + a\right )^{2} + b x - {\left (\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{6} - 4 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} - 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} - 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} - 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 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 ) + 8 \, {\left (b x \cosh \left (b x + a\right )^{7} - 3 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{5} + {\left (3 \, b x - 2\right )} \cosh \left (b x + a\right )^{3} - {\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{8} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} - 4 \, b \cosh \left (b x + a\right )^{6} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{6} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} - 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 6 \, b \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b \cosh \left (b x + a\right )^{4} - 30 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{5} - 10 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 4 \, b \cosh \left (b x + a\right )^{2} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{6} - 15 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (b \cosh \left (b x + a\right )^{7} - 3 \, b \cosh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 126 vs.
\(2 (32) = 64\).
time = 2.41, size = 126, normalized size = 3.00 \begin {gather*} \begin {cases} - \frac {\log {\left (- e^{- b x} \right )} \coth ^{5}{\left (b x + \log {\left (- e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {\log {\left (e^{- b x} \right )} \coth ^{5}{\left (b x + \log {\left (e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\x \coth ^{5}{\left (a \right )} & \text {for}\: b = 0 \\x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} + \frac {\log {\left (\tanh {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tanh ^{2}{\left (a + b x \right )}} - \frac {1}{4 b \tanh ^{4}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 70, normalized size = 1.67 \begin {gather*} -\frac {b x + a + \frac {4 \, {\left (e^{\left (6 \, b x + 6 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 159, normalized size = 3.79 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x-\frac {4}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {8}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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