Optimal. Leaf size=50 \[ -\frac{2 b (a+b) \tanh (c+d x)}{d}-\frac{(a+b)^2 \coth (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0622623, antiderivative size = 50, 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 = {4132, 270} \[ -\frac{2 b (a+b) \tanh (c+d x)}{d}-\frac{(a+b)^2 \coth (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 270
Rubi steps
\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^2}{x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 b (a+b)+\frac{(a+b)^2}{x^2}+b^2 x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^2 \coth (c+d x)}{d}-\frac{2 b (a+b) \tanh (c+d x)}{d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 1.52381, size = 109, normalized size = 2.18 \[ -\frac{4 \text{sech}^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (\sinh (d x) \cosh ^2(c+d x) \left (b (6 a+5 b) \text{sech}(c)-3 (a+b)^2 \text{csch}(c) \coth (c+d x)\right )+b^2 \tanh (c) \cosh (c+d x)+b^2 \text{sech}(c) \sinh (d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 91, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( -{a}^{2}{\rm coth} \left (dx+c\right )+2\,ab \left ( -{\frac{1}{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}-2\,\tanh \left ( dx+c \right ) \right ) +{b}^{2} \left ( -{\frac{1}{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-4\, \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06326, size = 189, normalized size = 3.78 \begin{align*} -\frac{16}{3} \, b^{2}{\left (\frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}} + \frac{1}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + \frac{2 \, a^{2}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac{8 \, a b}{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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Fricas [B] time = 2.57163, size = 733, normalized size = 14.66 \begin{align*} -\frac{4 \,{\left ({\left (3 \, a^{2} + 6 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} + 6 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - 2 \,{\left (3 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} +{\left (9 \, a^{2} + 18 \, a b + 8 \, b^{2}\right )} \cosh \left (d x + c\right ) - 2 \,{\left (3 \,{\left (3 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, a b + 4 \, 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 \operatorname{sech}^{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 [B] time = 1.22173, size = 151, normalized size = 3.02 \begin{align*} -\frac{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} + \frac{2 \,{\left (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 )} + 12 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b + 5 \, b^{2}\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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