Optimal. Leaf size=28 \[ -\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth (a+b x)}{b}+x \]
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Rubi [A] time = 0.02, 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 = {3473, 8} \[ -\frac {\coth ^3(a+b x)}{3 b}-\frac {\coth (a+b x)}{b}+x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
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] time = 0.01, size = 31, normalized size = 1.11 \[ -\frac {\coth ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 108, normalized size = 3.86 \[ \frac {{\left (3 \, b x + 4\right )} \sinh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )^{3} - 12 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, {\left ({\left (3 \, b x + 4\right )} \cosh \left (b x + a\right )^{2} - 3 \, b x - 4\right )} \sinh \left (b x + a\right )}{3 \, {\left (b \sinh \left (b x + a\right )^{3} + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 52, normalized size = 1.86 \[ \frac {3 \, b x + 3 \, a - \frac {4 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a\right )} + 2\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 54, normalized size = 1.93 \[ -\frac {\coth ^{3}\left (b x +a \right )}{3 b}-\frac {\coth \left (b x +a \right )}{b}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2 b}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 71, normalized size = 2.54 \[ x + \frac {a}{b} - \frac {4 \, {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 2\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \]
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 \[ x-\frac {\frac {{\mathrm {coth}\left (a+b\,x\right )}^3}{3}+\mathrm {coth}\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.49, size = 49, normalized size = 1.75 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = \log {\left (- e^{- b x} \right )} \vee a = \log {\left (e^{- b x} \right )} \\x \coth ^{4}{\relax (a )} & \text {for}\: b = 0 \\x - \frac {1}{b \tanh {\left (a + b x \right )}} - \frac {1}{3 b \tanh ^{3}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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