Optimal. Leaf size=33 \[ \frac{(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-b) \]
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Rubi [A] time = 0.0390378, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3675, 385, 206} \[ \frac{(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-b) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{1}{2} (a-b) x+\frac{(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.056917, size = 32, normalized size = 0.97 \[ \frac{2 (a-b) (c+d x)+(a+b) \sinh (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 54, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) +a \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11015, size = 93, normalized size = 2.82 \begin{align*} \frac{1}{8} \, a{\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{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86134, size = 80, normalized size = 2.42 \begin{align*} \frac{{\left (a - b\right )} d x +{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{2 \, d} \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 ) \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 = 1.21928, size = 109, normalized size = 3.3 \begin{align*} \frac{4 \,{\left (a - b\right )} d x -{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a e^{\left (2 \, d x + 4 \, c\right )} + b e^{\left (2 \, d x + 4 \, c\right )}\right )} e^{\left (-2 \, c\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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