Optimal. Leaf size=51 \[ \frac{(a+b)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-3 b) (a+b)+\frac{b^2 \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.0756621, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3675, 390, 385, 206} \[ \frac{(a+b)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-3 b) (a+b)+\frac{b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 390
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^2+\frac{a^2-b^2+2 b (a+b) x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^2 \tanh (c+d x)}{d}+\frac{\operatorname{Subst}\left (\int \frac{a^2-b^2+2 b (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b)^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^2 \tanh (c+d x)}{d}+\frac{((a-3 b) (a+b)) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{1}{2} (a-3 b) (a+b) x+\frac{(a+b)^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^2 \tanh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.370937, size = 54, normalized size = 1.06 \[ \frac{(a-3 b) (a+b) (c+d x)}{2 d}+\frac{(a+b)^2 \sinh (2 (c+d x))}{4 d}+\frac{b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 96, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,ab \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +{b}^{2} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{2\,\cosh \left ( dx+c \right ) }}-{\frac{3\,dx}{2}}-{\frac{3\,c}{2}}+{\frac{3\,\tanh \left ( dx+c \right ) }{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20452, size = 189, normalized size = 3.71 \begin{align*} \frac{1}{8} \, a^{2}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{4} \, a b{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{8} \, b^{2}{\left (\frac{12 \,{\left (d x + c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00785, size = 261, normalized size = 5.12 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} + 4 \,{\left ({\left (a^{2} - 2 \, a b - 3 \, b^{2}\right )} d x - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) +{\left (3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + 9 \, b^{2}\right )} \sinh \left (d x + c\right )}{8 \, d \cosh \left (d x + c\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 )^{2} \cosh ^{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 = 1.68859, size = 230, normalized size = 4.51 \begin{align*} \frac{4 \,{\left (a^{2} - 2 \, a b - 3 \, b^{2}\right )} d x +{\left (a^{2} e^{\left (2 \, d x + 8 \, c\right )} + 2 \, a b e^{\left (2 \, d x + 8 \, c\right )} + b^{2} e^{\left (2 \, d x + 8 \, c\right )}\right )} e^{\left (-6 \, c\right )} - \frac{{\left (a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-2 \, c\right )}}{e^{\left (2 \, d x\right )} + e^{\left (4 \, d x + 2 \, c\right )}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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