Optimal. Leaf size=30 \[ \frac{(a+b) \sinh (c+d x)}{d}+\frac{a \sinh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0502306, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4044, 3013} \[ \frac{(a+b) \sinh (c+d x)}{d}+\frac{a \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cosh ^3(c+d x) \left (a+b \text{sech}^2(c+d x)\right ) \, dx &=\int \cosh (c+d x) \left (b+a \cosh ^2(c+d x)\right ) \, dx\\ &=\frac{i \operatorname{Subst}\left (\int \left (a+b-a x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{d}\\ &=\frac{(a+b) \sinh (c+d x)}{d}+\frac{a \sinh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0173721, size = 50, normalized size = 1.67 \[ \frac{a \sinh ^3(c+d x)}{3 d}+\frac{a \sinh (c+d x)}{d}+\frac{b \sinh (c) \cosh (d x)}{d}+\frac{b \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( dx+c \right ) +b\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14902, size = 115, normalized size = 3.83 \begin{align*} \frac{1}{24} \, a{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} - \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac{1}{2} \, b{\left (\frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.099, size = 105, normalized size = 3.5 \begin{align*} \frac{a \sinh \left (d x + c\right )^{3} + 3 \,{\left (a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )}{12 \, 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 [B] time = 1.16692, size = 97, normalized size = 3.23 \begin{align*} \frac{a e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a e^{\left (d x + c\right )} + 12 \, b e^{\left (d x + c\right )} -{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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