Optimal. Leaf size=43 \[ x-\frac {\coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth ^5(a+b x)}{5 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8}
\begin {gather*} -\frac {\coth ^5(a+b x)}{5 b}-\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth (a+b x)}{b}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rubi steps
\begin {align*} \int \coth ^6(a+b x) \, dx &=-\frac {\coth ^5(a+b x)}{5 b}+\int \coth ^4(a+b x) \, dx\\ &=-\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth ^5(a+b x)}{5 b}+\int \coth ^2(a+b x) \, dx\\ &=-\frac {\coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth ^5(a+b x)}{5 b}+\int 1 \, dx\\ &=x-\frac {\coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 31, normalized size = 0.72 \begin {gather*} -\frac {\coth ^5(a+b x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\tanh ^2(a+b x)\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 56, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\coth ^{5}\left (b x +a \right )\right )}{5}-\frac {\left (\coth ^{3}\left (b x +a \right )\right )}{3}-\coth \left (b x +a \right )-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(56\) |
default | \(\frac {-\frac {\left (\coth ^{5}\left (b x +a \right )\right )}{5}-\frac {\left (\coth ^{3}\left (b x +a \right )\right )}{3}-\coth \left (b x +a \right )-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(56\) |
risch | \(x -\frac {2 \left (45 \,{\mathrm e}^{8 b x +8 a}-90 \,{\mathrm e}^{6 b x +6 a}+140 \,{\mathrm e}^{4 b x +4 a}-70 \,{\mathrm e}^{2 b x +2 a}+23\right )}{15 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{5}}\) | \(67\) |
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. 115 vs.
\(2 (39) = 78\).
time = 0.27, size = 115, normalized size = 2.67 \begin {gather*} x + \frac {a}{b} - \frac {2 \, {\left (70 \, e^{\left (-2 \, b x - 2 \, a\right )} - 140 \, e^{\left (-4 \, b x - 4 \, a\right )} + 90 \, e^{\left (-6 \, b x - 6 \, a\right )} - 45 \, e^{\left (-8 \, b x - 8 \, a\right )} - 23\right )}}{15 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, 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. 239 vs.
\(2 (39) = 78\).
time = 0.38, size = 239, normalized size = 5.56 \begin {gather*} \frac {{\left (15 \, b x + 23\right )} \sinh \left (b x + a\right )^{5} - 23 \, \cosh \left (b x + a\right )^{5} - 115 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 5 \, {\left (2 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{2} - 15 \, b x - 23\right )} \sinh \left (b x + a\right )^{3} + 25 \, \cosh \left (b x + a\right )^{3} - 5 \, {\left (46 \, \cosh \left (b x + a\right )^{3} - 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 5 \, {\left ({\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{4} - 3 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{2} + 30 \, b x + 46\right )} \sinh \left (b x + a\right ) - 50 \, \cosh \left (b x + a\right )}{15 \, {\left (b \sinh \left (b x + a\right )^{5} + 5 \, {\left (2 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{3} + 5 \, {\left (b \cosh \left (b x + a\right )^{4} - 3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )\right )}} \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. 114 vs.
\(2 (32) = 64\).
time = 3.85, size = 114, normalized size = 2.65 \begin {gather*} \begin {cases} - \frac {\log {\left (- e^{- b x} \right )} \coth ^{6}{\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 ^{6}{\left (b x + \log {\left (e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\x \coth ^{6}{\left (a \right )} & \text {for}\: b = 0 \\x - \frac {1}{b \tanh {\left (a + b x \right )}} - \frac {1}{3 b \tanh ^{3}{\left (a + b x \right )}} - \frac {1}{5 b \tanh ^{5}{\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 = 0.44, size = 74, normalized size = 1.72 \begin {gather*} \frac {15 \, b x + 15 \, a - \frac {2 \, {\left (45 \, e^{\left (8 \, b x + 8 \, a\right )} - 90 \, e^{\left (6 \, b x + 6 \, a\right )} + 140 \, e^{\left (4 \, b x + 4 \, a\right )} - 70 \, e^{\left (2 \, b x + 2 \, a\right )} + 23\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{5}}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 34, normalized size = 0.79 \begin {gather*} x-\frac {\frac {{\mathrm {coth}\left (a+b\,x\right )}^5}{5}+\frac {{\mathrm {coth}\left (a+b\,x\right )}^3}{3}+\mathrm {coth}\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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