Optimal. Leaf size=66 \[ \frac{(4 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}-\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.0455378, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3676, 385, 199, 203} \[ \frac{(4 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}-\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 385
Rule 199
Rule 203
Rubi steps
\begin{align*} \int \text{sech}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+(a+b) x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{b \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{(4 a+b) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac{(4 a+b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}-\frac{b \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{(4 a+b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac{(4 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(4 a+b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}-\frac{b \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0293792, size = 93, normalized size = 1.41 \[ \frac{a \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{8 d}-\frac{b \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 103, normalized size = 1.6 \begin{align*}{\frac{a{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{a\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}-{\frac{b\sinh \left ( dx+c \right ) }{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{b \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}\tanh \left ( dx+c \right ) }{12\,d}}+{\frac{b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61634, size = 244, normalized size = 3.7 \begin{align*} -\frac{1}{4} \, b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01207, size = 2826, normalized size = 42.82 \begin{align*} \text{result too large to display} \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}^{3}{\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 = 1.22584, size = 178, normalized size = 2.7 \begin{align*} \frac{{\left (4 \, a e^{c} + b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} + \frac{4 \, a e^{\left (7 \, d x + 7 \, c\right )} + b e^{\left (7 \, d x + 7 \, c\right )} + 4 \, a e^{\left (5 \, d x + 5 \, c\right )} - 7 \, b e^{\left (5 \, d x + 5 \, c\right )} - 4 \, a e^{\left (3 \, d x + 3 \, c\right )} + 7 \, b e^{\left (3 \, d x + 3 \, c\right )} - 4 \, a e^{\left (d x + c\right )} - b e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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