Optimal. Leaf size=40 \[ a^2 x+\frac{b (2 a+b) \tanh (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0341698, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4128, 390, 206} \[ a^2 x+\frac{b (2 a+b) \tanh (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4128
Rule 390
Rule 206
Rubi steps
\begin{align*} \int \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}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b (2 a+b)-b^2 x^2+\frac{a^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b (2 a+b) \tanh (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x+\frac{b (2 a+b) \tanh (c+d x)}{d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 0.420214, size = 106, normalized size = 2.65 \[ \frac{4 \text{sech}^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (3 a^2 d x \cosh ^3(c+d x)+2 b (3 a+b) \text{sech}(c) \sinh (d x) \cosh ^2(c+d x)+b^2 \tanh (c) \cosh (c+d x)+b^2 \text{sech}(c) \sinh (d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 47, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( dx+c \right ) +2\,ab\tanh \left ( dx+c \right ) +{b}^{2} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \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 = 1.11098, size = 162, normalized size = 4.05 \begin{align*} a^{2} x + \frac{4}{3} \, b^{2}{\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{4 \, a b}{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 = 2.00682, size = 447, normalized size = 11.18 \begin{align*} \frac{{\left (3 \, a^{2} d x - 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} d x - 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (3 \, a^{2} d x - 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) + 6 \,{\left ({\left (3 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a b + b^{2}\right )} \sinh \left (d x + c\right )}{3 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1386, size = 107, normalized size = 2.68 \begin{align*} \frac{{\left (d x + c\right )} a^{2}}{d} - \frac{4 \,{\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b + 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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