Optimal. Leaf size=49 \[ \frac{a^2 \log (\tanh (c+d x))}{d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0769667, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3670, 446, 72} \[ \frac{a^2 \log (\tanh (c+d x))}{d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{x \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2}{(1-x) x} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b^2-\frac{(a+b)^2}{-1+x}+\frac{a^2}{x}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{(a+b)^2 \log (\cosh (c+d x))}{d}+\frac{a^2 \log (\tanh (c+d x))}{d}-\frac{b^2 \tanh ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.125025, size = 48, normalized size = 0.98 \[ \frac{2 \left (a^2 \log (\tanh (c+d x))+(a+b)^2 \log (\cosh (c+d x))\right )-b^2 \tanh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 60, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{ab\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52864, size = 140, normalized size = 2.86 \begin{align*} b^{2}{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, 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{2 \, a b \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac{a^{2} \log \left (\sinh \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 = 2.09311, size = 1715, normalized size = 35. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \coth{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26452, size = 198, normalized size = 4.04 \begin{align*} \frac{a^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right )}{2 \, d} + \frac{{\left (2 \, a b + b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right )}{2 \, d} - \frac{2 \, a b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b^{2}{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 4 \, a b - 2 \, b^{2}}{2 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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