Optimal. Leaf size=56 \[ \frac{2 e^{4 c (a+b x)} \cosh (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt{\cosh ^2(a c+b c x)}} \]
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Rubi [A] time = 0.128711, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6720, 2282, 12, 264} \[ \frac{2 e^{4 c (a+b x)} \cosh (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt{\cosh ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 264
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\cosh ^2(a c+b c x)^{3/2}} \, dx &=\frac{\cosh (a c+b c x) \int e^{c (a+b x)} \text{sech}^3(a c+b c x) \, dx}{\sqrt{\cosh ^2(a c+b c x)}}\\ &=\frac{\cosh (a c+b c x) \operatorname{Subst}\left (\int \frac{8 x^3}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\cosh ^2(a c+b c x)}}\\ &=\frac{(8 \cosh (a c+b c x)) \operatorname{Subst}\left (\int \frac{x^3}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\cosh ^2(a c+b c x)}}\\ &=\frac{2 e^{4 c (a+b x)} \cosh (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt{\cosh ^2(a c+b c x)}}\\ \end{align*}
Mathematica [A] time = 0.0717623, size = 46, normalized size = 0.82 \[ \frac{4 e^{5 c (a+b x)} \sqrt{\cosh ^2(c (a+b x))}}{b c \left (e^{2 c (a+b x)}+1\right )^3} \]
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 = 1.06753, size = 113, normalized size = 2.02 \begin{align*} -\frac{4 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{b c{\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac{2}{b c{\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79339, size = 302, normalized size = 5.39 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )}}{b c \cosh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + b c \sinh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right ) +{\left (3 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\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 = 1.25059, size = 51, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{b c{\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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