Optimal. Leaf size=47 \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^2(c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0581107, antiderivative size = 47, 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} \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^2(c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d} \]
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 ^3(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^3\right )^2}{x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^2}+2 a b x+b^2 x^4\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{a^2 \coth (c+d x)}{d}+\frac{a b \tanh ^2(c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.220764, size = 94, normalized size = 2. \[ -\frac{a^2 \coth (c+d x)}{d}-\frac{a b \text{sech}^2(c+d x)}{d}+\frac{b^2 \tanh (c+d x)}{5 d}+\frac{b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}-\frac{2 b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 105, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ( -{a}^{2}{\rm coth} \left (dx+c\right )+{\frac{ab \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{b}^{2} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{3\,\tanh \left ( dx+c \right ) }{8} \left ({\frac{8}{15}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05468, size = 346, normalized size = 7.36 \begin{align*} \frac{2}{5} \, b^{2}{\left (\frac{10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac{5 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac{1}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac{2 \, a^{2}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} - \frac{4 \, a b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17751, size = 1334, normalized size = 28.38 \begin{align*} -\frac{4 \,{\left ({\left (5 \, a^{2} + 5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \,{\left (5 \, a^{2} + 5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} +{\left (5 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} +{\left (25 \, a^{2} + 5 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} +{\left (10 \,{\left (5 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 15 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} +{\left (10 \,{\left (5 \, a^{2} + 5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (25 \, a^{2} + 5 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \,{\left (5 \, a^{2} - a b\right )} \cosh \left (d x + c\right ) +{\left (5 \,{\left (5 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 9 \,{\left (5 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 10 \, a b + 10 \, b^{2}\right )} \sinh \left (d x + c\right )\right )}}{5 \,{\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{7} + 3 \, d \cosh \left (d x + c\right )^{5} +{\left (21 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{5} + 5 \,{\left (7 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + d \cosh \left (d x + c\right )^{3} +{\left (35 \, d \cosh \left (d x + c\right )^{4} + 50 \, d \cosh \left (d x + c\right )^{2} + 9 \, d\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (7 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 5 \, d \cosh \left (d x + c\right ) +{\left (7 \, d \cosh \left (d x + c\right )^{6} + 25 \, d \cosh \left (d x + c\right )^{4} + 27 \, d \cosh \left (d x + c\right )^{2} + 5 \, 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 ^{3}{\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 = 1.61257, size = 165, normalized size = 3.51 \begin{align*} -\frac{2 \,{\left (\frac{5 \, a^{2}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac{10 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 5 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 30 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 30 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 10 \, a b e^{\left (2 \, d x + 2 \, c\right )} + b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}\right )}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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