Optimal. Leaf size=54 \[ -\frac{3 x}{2 a}+\frac{4 \sinh ^3(x)}{3 a}+\frac{4 \sinh (x)}{a}-\frac{3 \sinh (x) \cosh (x)}{2 a}-\frac{\sinh (x) \cosh ^2(x)}{a \text{sech}(x)+a} \]
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Rubi [A] time = 0.0864547, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2633, 2635, 8} \[ -\frac{3 x}{2 a}+\frac{4 \sinh ^3(x)}{3 a}+\frac{4 \sinh (x)}{a}-\frac{3 \sinh (x) \cosh (x)}{2 a}-\frac{\sinh (x) \cosh ^2(x)}{a \text{sech}(x)+a} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\cosh ^2(x) \sinh (x)}{a+a \text{sech}(x)}-\frac{\int \cosh ^3(x) (-4 a+3 a \text{sech}(x)) \, dx}{a^2}\\ &=-\frac{\cosh ^2(x) \sinh (x)}{a+a \text{sech}(x)}-\frac{3 \int \cosh ^2(x) \, dx}{a}+\frac{4 \int \cosh ^3(x) \, dx}{a}\\ &=-\frac{3 \cosh (x) \sinh (x)}{2 a}-\frac{\cosh ^2(x) \sinh (x)}{a+a \text{sech}(x)}+\frac{(4 i) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{a}-\frac{3 \int 1 \, dx}{2 a}\\ &=-\frac{3 x}{2 a}+\frac{4 \sinh (x)}{a}-\frac{3 \cosh (x) \sinh (x)}{2 a}-\frac{\cosh ^2(x) \sinh (x)}{a+a \text{sech}(x)}+\frac{4 \sinh ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0728321, size = 53, normalized size = 0.98 \[ \frac{\text{sech}\left (\frac{x}{2}\right ) \left (45 \sinh \left (\frac{x}{2}\right )+18 \sinh \left (\frac{3 x}{2}\right )-2 \sinh \left (\frac{5 x}{2}\right )+\sinh \left (\frac{7 x}{2}\right )-36 x \cosh \left (\frac{x}{2}\right )\right )}{24 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 111, normalized size = 2.1 \begin{align*}{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{5}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{3}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{5}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16826, size = 89, normalized size = 1.65 \begin{align*} -\frac{3 \, x}{2 \, a} - \frac{21 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} - \frac{2 \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} - 69 \, e^{\left (-3 \, x\right )} - 1}{24 \,{\left (a e^{\left (-3 \, x\right )} + a e^{\left (-4 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.43033, size = 346, normalized size = 6.41 \begin{align*} \frac{\cosh \left (x\right )^{4} +{\left (4 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} +{\left (6 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 20\right )} \sinh \left (x\right )^{2} - 3 \,{\left (12 \, x - 1\right )} \cosh \left (x\right ) + 20 \, \cosh \left (x\right )^{2} +{\left (4 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} - 36 \, x + 32 \, \cosh \left (x\right ) + 39\right )} \sinh \left (x\right ) - 36 \, x - 69}{24 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\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*} \frac{\int \frac{\cosh ^{3}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14813, size = 95, normalized size = 1.76 \begin{align*} -\frac{3 \, x}{2 \, a} - \frac{{\left (69 \, e^{\left (3 \, x\right )} + 18 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, a{\left (e^{x} + 1\right )}} + \frac{a^{2} e^{\left (3 \, x\right )} - 3 \, a^{2} e^{\left (2 \, x\right )} + 21 \, a^{2} e^{x}}{24 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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