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 = 0.0161894, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 3473
Rule 8
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*}
Mathematica [C] time = 0.0098578, 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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Maple [B] time = 0.002, size = 54, normalized size = 1.9 \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{3}}{3\,b}}-{\frac{{\rm coth} \left (bx+a\right )}{b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )-1 \right ) }{2\,b}}+{\frac{\ln \left ({\rm coth} \left (bx+a\right )+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04297, size = 96, normalized size = 3.43 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11061, size = 286, normalized size = 10.21 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.25608, size = 49, normalized size = 1.75 \begin{align*} \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}{\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21794, size = 70, normalized size = 2.5 \begin{align*} \frac{b x + a}{b} - \frac{4 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a\right )} + 2\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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