Optimal. Leaf size=46 \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0567826, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3663, 270} \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 270
Rubi steps
\begin{align*} \int \text{csch}^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} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x^2}+b^2 x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.448521, size = 43, normalized size = 0.93 \[ \frac{b \tanh (c+d x) \left (6 a-b \text{sech}^2(c+d x)+b\right )-3 a^2 \coth (c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 68, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -{a}^{2}{\rm coth} \left (dx+c\right )+2\,ab\tanh \left ( dx+c \right ) +{b}^{2} \left ( -{\frac{\sinh \left ( dx+c \right ) }{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{\tanh \left ( dx+c \right ) }{2} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05402, size = 184, normalized size = 4. \begin{align*} \frac{2}{3} \, b^{2}{\left (\frac{3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac{1}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac{4 \, a b}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} + \frac{2 \, a^{2}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85856, size = 683, normalized size = 14.85 \begin{align*} -\frac{4 \,{\left ({\left (3 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} +{\left (9 \, a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) + 2 \,{\left (3 \,{\left (3 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, a b - b^{2}\right )} \sinh \left (d x + c\right )\right )}}{3 \,{\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5} + d \cosh \left (d x + c\right )^{3} +{\left (10 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{3} +{\left (10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 2 \, d \cosh \left (d x + c\right ) +{\left (5 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \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} \operatorname{csch}^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34978, size = 116, normalized size = 2.52 \begin{align*} -\frac{2 \,{\left (\frac{3 \, a^{2}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \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 )} + 6 \, a b + b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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