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 = 0.13, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 12
Rule 264
Rule 2282
Rule 6720
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*}
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Mathematica [A] time = 0.08, 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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fricas [B] time = 0.49, size = 120, normalized size = 2.14 \[ -\frac {2 \, {\left (3 \, \cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )}}{b c \cosh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + b c \sinh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right ) + {\left (3 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 38, normalized size = 0.68 \[ -\frac {2 \, {\left (2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {8 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (2 \cosh \left (2 b c x +2 a c \right )+2\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 84, normalized size = 1.50 \[ -\frac {4 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {2}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 76, normalized size = 1.36 \[ -\frac {4\,{\mathrm {e}}^{a\,c+b\,c\,x}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{b\,c\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int \frac {e^{b c x}}{\left (\cosh ^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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