Optimal. Leaf size=66 \[ -\frac {e^{-a-b x}}{8 b}-\frac {e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{40 b} \]
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Rubi [A] time = 0.05, antiderivative size = 66, 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, 448} \[ -\frac {e^{-a-b x}}{8 b}-\frac {e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{40 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 448
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2 \left (1+x^2\right )}{8 x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2 \left (1+x^2\right )}{x^2} \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}-x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{8 b}\\ &=-\frac {e^{-a-b x}}{8 b}-\frac {e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{40 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 51, normalized size = 0.77 \[ \frac {3 e^{-a-b x} \left (e^{6 (a+b x)}-5\right )-5 e^{a+b x} \left (e^{2 (a+b x)}+3\right )}{120 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 105, normalized size = 1.59 \[ -\frac {6 \, \cosh \left (b x + a\right )^{3} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \, \sinh \left (b x + a\right )^{3} - {\left (27 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right ) + 10 \, \cosh \left (b x + a\right )}{60 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 55, normalized size = 0.83 \[ \frac {{\left (3 \, e^{\left (5 \, b x + 10 \, a\right )} - 5 \, e^{\left (3 \, b x + 8 \, a\right )} - 15 \, e^{\left (b x + 6 \, a\right )}\right )} e^{\left (-5 \, a\right )} - 15 \, e^{\left (-b x - a\right )}}{120 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 69, normalized size = 1.05 \[ -\frac {\sinh \left (3 b x +3 a \right )}{24 b}+\frac {\sinh \left (5 b x +5 a \right )}{40 b}-\frac {\cosh \left (b x +a \right )}{4 b}-\frac {\cosh \left (3 b x +3 a \right )}{24 b}+\frac {\cosh \left (5 b x +5 a \right )}{40 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 53, normalized size = 0.80 \[ -\frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 15 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3\right )} e^{\left (5 \, b x + 5 \, a\right )}}{120 \, b} - \frac {e^{\left (-b x - a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 47, normalized size = 0.71 \[ -\frac {15\,{\mathrm {e}}^{a+b\,x}+15\,{\mathrm {e}}^{-a-b\,x}+5\,{\mathrm {e}}^{3\,a+3\,b\,x}-3\,{\mathrm {e}}^{5\,a+5\,b\,x}}{120\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.53, size = 128, normalized size = 1.94 \[ \begin {cases} \frac {e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )}}{15 b} - \frac {2 e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b} + \frac {8 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{15 b} - \frac {4 e^{2 a} e^{2 b x} \cosh ^{3}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{2}{\relax (a )} \cosh {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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