Optimal. Leaf size=32 \[ \frac{a \tanh (c+d x)}{d}-\frac{(a-b) \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0337702, antiderivative size = 32, 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 = {3191} \[ \frac{a \tanh (c+d x)}{d}-\frac{(a-b) \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3191
Rubi steps
\begin{align*} \int \text{sech}^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a-(a-b) x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a \tanh (c+d x)}{d}-\frac{(a-b) \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0074434, size = 44, normalized size = 1.38 \[ -\frac{a \tanh ^3(c+d x)}{3 d}+\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 65, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) \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 [B] time = 1.0453, size = 250, normalized size = 7.81 \begin{align*} \frac{4}{3} \, a{\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}{3} \, b{\left (\frac{3 \, e^{\left (-4 \, d x - 4 \, 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38691, size = 427, normalized size = 13.34 \begin{align*} -\frac{4 \,{\left ({\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} - 2 \,{\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (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(-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 [A] time = 1.18553, size = 63, normalized size = 1.97 \begin{align*} -\frac{2 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, 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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