Optimal. Leaf size=19 \[ -\frac {1}{6} e^{-3 x}+\frac {e^{5 x}}{10} \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 12, 14}
\begin {gather*} \frac {e^{5 x}}{10}-\frac {1}{6} e^{-3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2320
Rubi steps
\begin {align*} \int e^x \cosh (4 x) \, dx &=\text {Subst}\left (\int \frac {1+x^8}{2 x^4} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1+x^8}{x^4} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^4}+x^4\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{6} e^{-3 x}+\frac {e^{5 x}}{10}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {1}{6} e^{-3 x}+\frac {e^{5 x}}{10} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 26, normalized size = 1.37
method | result | size |
risch | \(\frac {{\mathrm e}^{5 x}}{10}-\frac {{\mathrm e}^{-3 x}}{6}\) | \(14\) |
default | \(\frac {\sinh \left (3 x \right )}{6}+\frac {\sinh \left (5 x \right )}{10}-\frac {\cosh \left (3 x \right )}{6}+\frac {\cosh \left (5 x \right )}{10}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{10} \, e^{\left (5 \, x\right )} - \frac {1}{6} \, e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (13) = 26\).
time = 0.36, size = 46, normalized size = 2.42 \begin {gather*} -\frac {\cosh \left (x\right )^{4} - 16 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}{15 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 20, normalized size = 1.05 \begin {gather*} \frac {4 e^{x} \sinh {\left (4 x \right )}}{15} - \frac {e^{x} \cosh {\left (4 x \right )}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{10} \, e^{\left (5 \, x\right )} - \frac {1}{6} \, e^{\left (-3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 14, normalized size = 0.74 \begin {gather*} \frac {{\mathrm {e}}^{-3\,x}\,\left (3\,{\mathrm {e}}^{8\,x}-5\right )}{30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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