Optimal. Leaf size=63 \[ \frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{(3 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (3 a-b) \]
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Rubi [A] time = 0.0496557, antiderivative size = 63, 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 = {3675, 385, 199, 206} \[ \frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac{(3 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x (3 a-b) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 385
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \cosh ^4(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 )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b) \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{(3 a-b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac{(3 a-b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{(a+b) \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{(3 a-b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{1}{8} (3 a-b) x+\frac{(3 a-b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{(a+b) \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.162677, size = 44, normalized size = 0.7 \[ \frac{(a+b) \sinh (4 (c+d x))+12 a (c+d x)+8 a \sinh (2 (c+d x))-4 b d x}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 82, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{8}}-{\frac{dx}{8}}-{\frac{c}{8}} \right ) +a \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14363, size = 140, normalized size = 2.22 \begin{align*} \frac{1}{64} \, a{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac{1}{64} \, b{\left (\frac{8 \,{\left (d x + c\right )}}{d} - \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82498, size = 169, normalized size = 2.68 \begin{align*} \frac{{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (3 \, a - b\right )} d x +{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 4 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \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 = 1.27921, size = 144, normalized size = 2.29 \begin{align*} \frac{8 \,{\left (3 \, a - b\right )} d x -{\left (18 \, a e^{\left (4 \, d x + 4 \, c\right )} - 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} +{\left (a e^{\left (4 \, d x + 12 \, c\right )} + b e^{\left (4 \, d x + 12 \, c\right )} + 8 \, a e^{\left (2 \, d x + 10 \, c\right )}\right )} e^{\left (-8 \, c\right )}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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