Optimal. Leaf size=49 \[ -\frac {e^{-3 a-3 b x}}{48 b}-\frac {e^{a+b x}}{8 b}+\frac {e^{5 a+5 b x}}{80 b} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2282, 12, 270} \[ -\frac {e^{-3 a-3 b x}}{48 b}-\frac {e^{a+b x}}{8 b}+\frac {e^{5 a+5 b x}}{80 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2282
Rubi steps
\begin {align*} \int e^{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^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^2}{x^4} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x^4}+x^4\right ) \, dx,x,e^{a+b x}\right )}{16 b}\\ &=-\frac {e^{-3 a-3 b x}}{48 b}-\frac {e^{a+b x}}{8 b}+\frac {e^{5 a+5 b x}}{80 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.82 \[ \frac {e^{-3 (a+b x)} \left (-30 e^{4 (a+b x)}+3 e^{8 (a+b x)}-5\right )}{240 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 90, normalized size = 1.84 \[ -\frac {\cosh \left (b x + a\right )^{4} - 16 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 15}{120 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 36, normalized size = 0.73 \[ \frac {3 \, e^{\left (5 \, b x + 5 \, a\right )} - 30 \, e^{\left (b x + a\right )} - 5 \, e^{\left (-3 \, b x - 3 \, a\right )}}{240 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 70, normalized size = 1.43 \[ \frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{15}+\frac {\sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{5}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 38, normalized size = 0.78 \[ \frac {e^{\left (5 \, b x + 5 \, a\right )} - 10 \, e^{\left (b x + a\right )}}{80 \, b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 36, normalized size = 0.73 \[ -\frac {30\,{\mathrm {e}}^{a+b\,x}+5\,{\mathrm {e}}^{-3\,a-3\,b\,x}-3\,{\mathrm {e}}^{5\,a+5\,b\,x}}{240\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 62.53, size = 144, normalized size = 2.94 \[ \begin {cases} - \frac {2 e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )}}{15 b} + \frac {2 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b} + \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} + \frac {2 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{15 b} - \frac {2 e^{a} e^{b x} \cosh ^{4}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{2}{\relax (a )} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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