Optimal. Leaf size=36 \[ -\frac{a^2 \coth (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.0671894, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 461, 207} \[ -\frac{a^2 \coth (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 207
Rubi steps
\begin{align*} \int \coth ^2(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^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b^2+\frac{a^2}{x^2}-\frac{(a+b)^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a^2 \coth (c+d x)}{d}-\frac{b^2 \tanh (c+d x)}{d}-\frac{(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac{a^2 \coth (c+d x)}{d}-\frac{b^2 \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 0.0959733, size = 64, normalized size = 1.78 \[ -\frac{a^2 \coth (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right )}{d}+2 a b x+\frac{b^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 49, normalized size = 1.4 \begin{align*}{\frac{{a}^{2} \left ( dx+c-{\rm coth} \left (dx+c\right ) \right ) +2\, \left ( dx+c \right ) ab+{b}^{2} \left ( dx+c-\tanh \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07441, size = 86, normalized size = 2.39 \begin{align*} b^{2}{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{2}{\left (x + \frac{c}{d} + \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + 2 \, a b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90056, size = 243, normalized size = 6.75 \begin{align*} -\frac{{\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - 2 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} d x + a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - b^{2}}{2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )} \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 = 1.30827, size = 99, normalized size = 2.75 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (d x + c\right )}}{d} - \frac{2 \,{\left (a^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + b^{2}\right )}}{d{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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