Optimal. Leaf size=35 \[ \frac {e^{-a-b x}}{4 b}+\frac {e^{3 a+3 b x}}{12 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2320, 12, 14}
\begin {gather*} \frac {e^{-a-b x}}{4 b}+\frac {e^{3 a+3 b x}}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \cosh (a+b x) \sinh (a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {-1+x^4}{4 x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {-1+x^4}{x^2} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x^2}+x^2\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {e^{-a-b x}}{4 b}+\frac {e^{3 a+3 b x}}{12 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} \frac {e^{-a-b x} \left (3+e^{4 (a+b x)}\right )}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 26, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sinh ^{3}\left (b x +a \right )\right )}{3}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(26\) |
default | \(\frac {\frac {\left (\sinh ^{3}\left (b x +a \right )\right )}{3}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(26\) |
risch | \(\frac {{\mathrm e}^{-b x -a}}{4 b}+\frac {{\mathrm e}^{3 b x +3 a}}{12 b}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 0.83 \begin {gather*} \frac {e^{\left (3 \, b x + 3 \, a\right )}}{12 \, b} + \frac {e^{\left (-b x - a\right )}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 53, normalized size = 1.51 \begin {gather*} \frac {\cosh \left (b x + a\right )^{2} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}}{3 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (24) = 48\).
time = 0.47, size = 76, normalized size = 2.17 \begin {gather*} \begin {cases} \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )}}{3 b} - \frac {e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b} + \frac {e^{a} e^{b x} \cosh ^{2}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh {\left (a \right )} \cosh {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 26, normalized size = 0.74 \begin {gather*} \frac {e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (-b x - a\right )}}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.75, size = 26, normalized size = 0.74 \begin {gather*} \frac {3\,{\mathrm {e}}^{-a-b\,x}+{\mathrm {e}}^{3\,a+3\,b\,x}}{12\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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