Optimal. Leaf size=60 \[ \frac{(a+b)^2 \sinh (c+d x)}{d}-\frac{b (4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b^2 \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.0824405, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3676, 390, 385, 203} \[ \frac{(a+b)^2 \sinh (c+d x)}{d}-\frac{b (4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b^2 \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 390
Rule 385
Rule 203
Rubi steps
\begin{align*} \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+(a+b) x^2\right )^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left ((a+b)^2-\frac{b (2 a+b)+2 b (a+b) x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a+b)^2 \sinh (c+d x)}{d}-\frac{\operatorname{Subst}\left (\int \frac{b (2 a+b)+2 b (a+b) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a+b)^2 \sinh (c+d x)}{d}+\frac{b^2 \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{(b (4 a+3 b)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=-\frac{b (4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a+b)^2 \sinh (c+d x)}{d}+\frac{b^2 \text{sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.178763, size = 54, normalized size = 0.9 \[ \frac{2 (a+b)^2 \sinh (c+d x)+b \left (b \tanh (c+d x) \text{sech}(c+d x)-(4 a+3 b) \tan ^{-1}(\sinh (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 122, normalized size = 2. \begin{align*}{\frac{{a}^{2}\sinh \left ( dx+c \right ) }{d}}+2\,{\frac{ab\sinh \left ( dx+c \right ) }{d}}-4\,{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}-3\,{\frac{{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.74926, size = 205, normalized size = 3.42 \begin{align*} \frac{1}{2} \, b^{2}{\left (\frac{6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )}}{d} + \frac{4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d{\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + a b{\left (\frac{4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a^{2} \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99283, size = 1967, normalized size = 32.78 \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 )^{2} \cosh{\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 = 1.66563, size = 185, normalized size = 3.08 \begin{align*} -\frac{2 \,{\left (4 \, a b e^{c} + 3 \, b^{2} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} +{\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-d x - c\right )} -{\left (a^{2} e^{\left (d x + 8 \, c\right )} + 2 \, a b e^{\left (d x + 8 \, c\right )} + b^{2} e^{\left (d x + 8 \, c\right )}\right )} e^{\left (-7 \, c\right )} - \frac{2 \,{\left (b^{2} e^{\left (3 \, d x + 3 \, c\right )} - b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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