Optimal. Leaf size=28 \[ \frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0293381, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3675} \[ \frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rubi steps
\begin{align*} \int \text{sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0112985, size = 28, normalized size = 1. \[ \frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 53, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( a\tanh \left ( dx+c \right ) +b \left ( -{\frac{\sinh \left ( dx+c \right ) }{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{\tanh \left ( dx+c \right ) }{2} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13298, size = 46, normalized size = 1.64 \begin{align*} \frac{b \tanh \left (d x + c\right )^{3}}{3 \, d} + \frac{2 \, a}{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 = 1.81801, size = 424, normalized size = 15.14 \begin{align*} -\frac{4 \,{\left ({\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (3 \, a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 3 \, a\right )}}{3 \,{\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 3 \, 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 \tanh ^{2}{\left (c + d x \right )}\right ) \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 = 1.1891, size = 80, normalized size = 2.86 \begin{align*} -\frac{2 \,{\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a + 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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