Optimal. Leaf size=162 \[ -\frac{e^{-2 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{16 b c}+\frac{3 e^{2 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{16 b c}+\frac{e^{4 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{32 b c}+\frac{3}{8} x \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x) \]
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Rubi [A] time = 0.123512, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 2282, 12, 266, 43} \[ -\frac{e^{-2 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{16 b c}+\frac{3 e^{2 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{16 b c}+\frac{e^{4 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{32 b c}+\frac{3}{8} x \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x) \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{c (a+b x)} \cosh ^2(a c+b c x)^{3/2} \, dx &=\left (\sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)\right ) \int e^{c (a+b x)} \cosh ^3(a c+b c x) \, dx\\ &=\frac{\left (\sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{8 x^3} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (\sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^3} \, dx,x,e^{c (a+b x)}\right )}{8 b c}\\ &=\frac{\left (\sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{(1+x)^3}{x^2} \, dx,x,e^{2 c (a+b x)}\right )}{16 b c}\\ &=\frac{\left (\sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)\right ) \operatorname{Subst}\left (\int \left (3+\frac{1}{x^2}+\frac{3}{x}+x\right ) \, dx,x,e^{2 c (a+b x)}\right )}{16 b c}\\ &=-\frac{e^{-2 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{16 b c}+\frac{3 e^{2 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{16 b c}+\frac{e^{4 c (a+b x)} \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)}{32 b c}+\frac{3}{8} x \sqrt{\cosh ^2(a c+b c x)} \text{sech}(a c+b c x)\\ \end{align*}
Mathematica [A] time = 0.107306, size = 78, normalized size = 0.48 \[ \frac{\left (-e^{-2 c (a+b x)}+3 e^{2 c (a+b x)}+\frac{1}{2} e^{4 c (a+b x)}+6 b c x\right ) \cosh ^2(c (a+b x))^{3/2} \text{sech}^3(c (a+b x))}{16 b c} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c \left ( bx+a \right ) }} \left ( \left ( \cosh \left ( bcx+ac \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15058, size = 100, normalized size = 0.62 \begin{align*} \frac{3 \,{\left (b c x + a c\right )}}{8 \, b c} + \frac{e^{\left (4 \, b c x + 4 \, a c\right )}}{32 \, b c} + \frac{3 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{16 \, b c} - \frac{e^{\left (-2 \, b c x - 2 \, a c\right )}}{16 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01751, size = 323, normalized size = 1.99 \begin{align*} -\frac{\cosh \left (b c x + a c\right )^{3} + 3 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} - 3 \, \sinh \left (b c x + a c\right )^{3} - 6 \,{\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) + 3 \,{\left (4 \, b c x - 3 \, \cosh \left (b c x + a c\right )^{2} - 2\right )} \sinh \left (b c x + a c\right )}{32 \,{\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \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 = 1.2913, size = 99, normalized size = 0.61 \begin{align*} \frac{12 \, b c x - 2 \,{\left (3 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )} e^{\left (-2 \, b c x - 2 \, a c\right )} +{\left (e^{\left (4 \, b c x + 8 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 6 \, a c\right )}\right )} e^{\left (-4 \, a c\right )}}{32 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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