Optimal. Leaf size=45 \[ -\frac{(a+b) \coth ^3(c+d x)}{3 d}+\frac{(a+2 b) \coth (c+d x)}{d}+\frac{b \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.0579728, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4132, 448} \[ -\frac{(a+b) \coth ^3(c+d x)}{3 d}+\frac{(a+2 b) \coth (c+d x)}{d}+\frac{b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 448
Rubi steps
\begin{align*} \int \text{csch}^4(c+d x) \left (a+b \text{sech}^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (a+b-b x^2\right )}{x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b+\frac{a+b}{x^4}+\frac{-a-2 b}{x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+2 b) \coth (c+d x)}{d}-\frac{(a+b) \coth ^3(c+d x)}{3 d}+\frac{b \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.05242, size = 84, normalized size = 1.87 \[ \frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{b \tanh (c+d x)}{d}+\frac{5 b \coth (c+d x)}{3 d}-\frac{b \coth (c+d x) \text{csch}^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 73, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right )+b \left ( -{\frac{1}{3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}\cosh \left ( dx+c \right ) }}+{\frac{4}{3\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}+{\frac{8\,\tanh \left ( dx+c \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04763, size = 252, normalized size = 5.6 \begin{align*} \frac{4}{3} \, a{\left (\frac{3 \, e^{\left (-2 \, d x - 2 \, 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{16}{3} \, b{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.46678, size = 666, normalized size = 14.8 \begin{align*} -\frac{8 \,{\left ({\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + a + 4 \, b\right )}}{3 \,{\left (d \cosh \left (d x + c\right )^{6} + 6 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + d \sinh \left (d x + c\right )^{6} - 2 \, d \cosh \left (d x + c\right )^{4} +{\left (15 \, d \cosh \left (d x + c\right )^{2} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, d \cosh \left (d x + c\right )^{3} - 2 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )^{2} +{\left (15 \, d \cosh \left (d x + c\right )^{4} - 12 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{5} - 4 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2 \, d\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 ) \operatorname{csch}^{4}{\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 = 1.15938, size = 109, normalized size = 2.42 \begin{align*} -\frac{2 \, b}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} + \frac{2 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 5 \, b\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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