Optimal. Leaf size=42 \[ -\frac {\coth ^4(a+b x)}{4 b}-\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \]
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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 = {3473, 3475} \[ -\frac {\coth ^4(a+b x)}{4 b}-\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
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 = 0.18, size = 44, normalized size = 1.05 \[ -\frac {\coth ^4(a+b x)+2 \coth ^2(a+b x)-4 \log (\tanh (a+b x))-4 \log (\cosh (a+b x))}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 978, normalized size = 23.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 70, normalized size = 1.67 \[ -\frac {b x + a + \frac {4 \, {\left (e^{\left (6 \, b x + 6 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 1.33 \[ -\frac {\coth ^{4}\left (b x +a \right )}{4 b}-\frac {\coth ^{2}\left (b x +a \right )}{2 b}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2 b}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.05, size = 122, normalized size = 2.90 \[ x + \frac {a}{b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \]
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 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x-\frac {4}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {8}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.17, size = 75, normalized size = 1.79 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = \log {\left (- e^{- b x} \right )} \vee a = \log {\left (e^{- b x} \right )} \\x \coth ^{5}{\relax (a )} & \text {for}\: b = 0 \\x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} + \frac {\log {\left (\tanh {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tanh ^{2}{\left (a + b x \right )}} - \frac {1}{4 b \tanh ^{4}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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