Optimal. Leaf size=53 \[ -\frac{2 b (a+b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \tanh (c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0656931, antiderivative size = 53, 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 = {4146, 194} \[ -\frac{2 b (a+b) \tanh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \tanh (c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 194
Rubi steps
\begin{align*} \int \text{sech}^2(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b-b x^2\right )^2 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1+\frac{b (2 a+b)}{a^2}\right )-2 a b \left (1+\frac{b}{a}\right ) x^2+b^2 x^4\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b)^2 \tanh (c+d x)}{d}-\frac{2 b (a+b) \tanh ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0295908, size = 93, normalized size = 1.75 \[ \frac{a^2 \tanh (c+d x)}{d}-\frac{2 a b \tanh ^3(c+d x)}{3 d}+\frac{2 a b \tanh (c+d x)}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}-\frac{2 b^2 \tanh ^3(c+d x)}{3 d}+\frac{b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 70, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{2}\tanh \left ( dx+c \right ) +2\,ab \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) +{b}^{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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11952, size = 437, normalized size = 8.25 \begin{align*} \frac{16}{15} \, b^{2}{\left (\frac{5 \, e^{\left (-2 \, d x - 2 \, 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{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{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{8}{3} \, a b{\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{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 = 2.07396, size = 1054, normalized size = 19.89 \begin{align*} -\frac{4 \,{\left ({\left (15 \, a^{2} + 10 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} - 8 \,{\left (5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (15 \, a^{2} + 10 \, a b + 4 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 20 \,{\left (3 \, a^{2} + 4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (15 \, a^{2} + 10 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 30 \, a^{2} + 40 \, a b + 10 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + 45 \, a^{2} + 70 \, a b + 40 \, b^{2} - 8 \,{\left ({\left (5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 5 \,{\left (a b + b^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname{sech}^{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 = 1.1662, size = 211, normalized size = 3.98 \begin{align*} -\frac{2 \,{\left (15 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 60 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 140 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 100 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 40 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} + 20 \, a b + 8 \, 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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