Optimal. Leaf size=28 \[ x-\frac {\coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, 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, 8}
\begin {gather*} -\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 ^4(a+b x) \, dx &=-\frac {\coth ^3(a+b x)}{3 b}+\int \coth ^2(a+b x) \, dx\\ &=-\frac {\coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x-\frac {\coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 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 = 1.11 \begin {gather*} -\frac {\coth ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(a+b x)\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 46, normalized size = 1.64
method | result | size |
risch | \(x -\frac {4 \left (3 \,{\mathrm e}^{4 b x +4 a}-3 \,{\mathrm e}^{2 b x +2 a}+2\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{3}}\) | \(45\) |
derivativedivides | \(\frac {-\frac {\left (\coth ^{3}\left (b x +a \right )\right )}{3}-\coth \left (b x +a \right )-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(46\) |
default | \(\frac {-\frac {\left (\coth ^{3}\left (b x +a \right )\right )}{3}-\coth \left (b x +a \right )-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(46\) |
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. 71 vs.
\(2 (26) = 52\).
time = 0.26, size = 71, normalized size = 2.54 \begin {gather*} x + \frac {a}{b} - \frac {4 \, {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 2\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, 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. 108 vs.
\(2 (26) = 52\).
time = 0.36, size = 108, normalized size = 3.86 \begin {gather*} \frac {{\left (3 \, b x + 4\right )} \sinh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )^{3} - 12 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, {\left ({\left (3 \, b x + 4\right )} \cosh \left (b x + a\right )^{2} - 3 \, b x - 4\right )} \sinh \left (b x + a\right )}{3 \, {\left (b \sinh \left (b x + a\right )^{3} + 3 \, {\left (b \cosh \left (b x + a\right )^{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. 100 vs.
\(2 (20) = 40\).
time = 1.38, size = 100, normalized size = 3.57 \begin {gather*} \begin {cases} - \frac {\log {\left (- e^{- b x} \right )} \coth ^{4}{\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 ^{4}{\left (b x + \log {\left (e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\x \coth ^{4}{\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 )}} & \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 = 52, normalized size = 1.86 \begin {gather*} \frac {3 \, b x + 3 \, a - \frac {4 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a\right )} + 2\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 24, normalized size = 0.86 \begin {gather*} x-\frac {\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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