Optimal. Leaf size=31 \[ \frac{(a+b) \log (\sinh (c+d x))}{d}-\frac{a \coth ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0424706, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3629, 12, 3475} \[ \frac{(a+b) \log (\sinh (c+d x))}{d}-\frac{a \coth ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3629
Rule 12
Rule 3475
Rubi steps
\begin{align*} \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac{a \coth ^2(c+d x)}{2 d}+\int (a+b) \coth (c+d x) \, dx\\ &=-\frac{a \coth ^2(c+d x)}{2 d}+(a+b) \int \coth (c+d x) \, dx\\ &=-\frac{a \coth ^2(c+d x)}{2 d}+\frac{(a+b) \log (\sinh (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.103553, size = 39, normalized size = 1.26 \[ \frac{2 (a+b) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))-a \coth ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 40, normalized size = 1.3 \begin{align*}{\frac{a\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}a}{2\,d}}+{\frac{b\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0785, size = 143, normalized size = 4.61 \begin{align*} a{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac{b \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10567, size = 1114, normalized size = 35.94 \begin{align*} -\frac{{\left (a + b\right )} d x \cosh \left (d x + c\right )^{4} + 4 \,{\left (a + b\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a + b\right )} d x \sinh \left (d x + c\right )^{4} +{\left (a + b\right )} d x - 2 \,{\left ({\left (a + b\right )} d x - a\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a + b\right )} d x \cosh \left (d x + c\right )^{2} -{\left (a + b\right )} d x + a\right )} \sinh \left (d x + c\right )^{2} -{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \,{\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} -{\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b\right )} \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \,{\left ({\left (a + b\right )} d x \cosh \left (d x + c\right )^{3} -{\left ({\left (a + b\right )} d x - a\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18364, size = 84, normalized size = 2.71 \begin{align*} -\frac{{\left (d x + c\right )}{\left (a + b\right )}}{d} + \frac{{\left (a + b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{d} - \frac{2 \, a e^{\left (2 \, d x + 2 \, c\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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