Optimal. Leaf size=57 \[ \frac{a^2 \tanh (c+d x)}{d}+\frac{(a-b)^2 \tanh ^5(c+d x)}{5 d}-\frac{2 a (a-b) \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0601492, antiderivative size = 57, 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 = {3191, 194} \[ \frac{a^2 \tanh (c+d x)}{d}+\frac{(a-b)^2 \tanh ^5(c+d x)}{5 d}-\frac{2 a (a-b) \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 194
Rubi steps
\begin{align*} \int \text{sech}^6(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a-(a-b) x^2\right )^2 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2-2 a (a-b) x^2+(a-b)^2 x^4\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a^2 \tanh (c+d x)}{d}-\frac{2 a (a-b) \tanh ^3(c+d x)}{3 d}+\frac{(a-b)^2 \tanh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.373003, size = 69, normalized size = 1.21 \[ \frac{\tanh (c+d x) \left (2 \left (2 a^2+a b-3 b^2\right ) \text{sech}^2(c+d x)+8 a^2+3 (a-b)^2 \text{sech}^4(c+d x)+4 a b+3 b^2\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 158, normalized size = 2.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \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 ) \tanh \left ( dx+c \right ) +2\,ab \left ( -1/4\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+1/4\, \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) +{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 = 1.12718, size = 942, normalized size = 16.53 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45971, size = 1040, normalized size = 18.25 \begin{align*} -\frac{4 \,{\left ({\left (4 \, a^{2} + 2 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} - 8 \,{\left (2 \, a^{2} + a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (4 \, a^{2} + 2 \, a b + 9 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 20 \,{\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (4 \, a^{2} + 2 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 10 \, a^{2} + 20 \, a b\right )} \sinh \left (d x + c\right )^{2} + 40 \, a^{2} - 10 \, a b + 15 \, b^{2} - 8 \,{\left ({\left (2 \, a^{2} + a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 5 \,{\left (a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{15 \,{\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} + 6 \, d \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, 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} + 4 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 15 \, d \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, d \cosh \left (d x + c\right )^{4} + 12 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{5} + 8 \, d \cosh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 10 \, d\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 = 1.22235, size = 173, normalized size = 3.04 \begin{align*} -\frac{2 \,{\left (15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 20 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 30 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 20 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} + 4 \, a b + 3 \, b^{2}\right )}}{15 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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