Optimal. Leaf size=52 \[ -\frac {e^{-2 a-2 b x}}{32 b}-\frac {e^{2 a+2 b x}}{16 b}+\frac {e^{6 a+6 b x}}{96 b} \]
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Rubi [A] time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2282, 12, 270} \[ -\frac {e^{-2 a-2 b x}}{32 b}-\frac {e^{2 a+2 b x}}{16 b}+\frac {e^{6 a+6 b x}}{96 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^2}{16 x^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^2}{x^3} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^3}-2 x+x^5\right ) \, dx,x,e^{a+b x}\right )}{16 b}\\ &=-\frac {e^{-2 a-2 b x}}{32 b}-\frac {e^{2 a+2 b x}}{16 b}+\frac {e^{6 a+6 b x}}{96 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 0.73 \[ \frac {e^{-2 (a+b x)} \left (-6 e^{4 (a+b x)}+e^{8 (a+b x)}-3\right )}{96 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 108, normalized size = 2.08 \[ -\frac {\cosh \left (b x + a\right )^{4} - 8 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 3}{48 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 43, normalized size = 0.83 \[ \frac {{\left (e^{\left (6 \, b x + 12 \, a\right )} - 6 \, e^{\left (2 \, b x + 8 \, a\right )}\right )} e^{\left (-6 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )}}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 58, normalized size = 1.12 \[ -\frac {\sinh \left (2 b x +2 a \right )}{32 b}+\frac {\sinh \left (6 b x +6 a \right )}{96 b}-\frac {3 \cosh \left (2 b x +2 a \right )}{32 b}+\frac {\cosh \left (6 b x +6 a \right )}{96 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 42, normalized size = 0.81 \[ -\frac {{\left (6 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{96 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 39, normalized size = 0.75 \[ -\frac {3\,{\mathrm {e}}^{-2\,a-2\,b\,x}+6\,{\mathrm {e}}^{2\,a+2\,b\,x}-{\mathrm {e}}^{6\,a+6\,b\,x}}{96\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 57.10, size = 128, normalized size = 2.46 \[ \begin {cases} - \frac {5 e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )}}{48 b} + \frac {5 e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{24 b} + \frac {e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac {e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{2}{\relax (a )} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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