Optimal. Leaf size=53 \[ \frac{(a+b)^2 \log (\sinh (c+d x))}{d}-\frac{b (2 a+b) \log (\cosh (c+d x))}{d}+\frac{b^2 \text{sech}^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0801266, antiderivative size = 53, 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 = {4138, 446, 88} \[ \frac{(a+b)^2 \log (\sinh (c+d x))}{d}-\frac{b (2 a+b) \log (\cosh (c+d x))}{d}+\frac{b^2 \text{sech}^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 4138
Rule 446
Rule 88
Rubi steps
\begin{align*} \int \coth (c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (b+a x^2\right )^2}{x^3 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^2}{(1-x) x^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^2}{-1+x}+\frac{b^2}{x^2}+\frac{b (2 a+b)}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{b (2 a+b) \log (\cosh (c+d x))}{d}+\frac{(a+b)^2 \log (\sinh (c+d x))}{d}+\frac{b^2 \text{sech}^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.242977, size = 84, normalized size = 1.58 \[ \frac{2 (a \cosh (c+d x)+b \text{sech}(c+d x))^2 \left (2 \cosh ^2(c+d x) \left ((a+b)^2 \log (\sinh (c+d x))-b (2 a+b) \log (\cosh (c+d x))\right )+b^2\right )}{d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 60, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{ab\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( \tanh \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 = 1.86205, size = 217, normalized size = 4.09 \begin{align*} b^{2}{\left (\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{\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 )} + 2 \, a b{\left (\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{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \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 = 2.21251, size = 1709, normalized size = 32.25 \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(-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 = 1.18386, size = 231, normalized size = 4.36 \begin{align*} -\frac{2 \, a^{2} d x + 2 \,{\left (2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 2 \,{\left (a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) - \frac{6 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 10 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b + 3 \, b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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