Optimal. Leaf size=19 \[ -\frac {1}{4} e^{-2 x}+\frac {e^{4 x}}{8} \]
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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^{4 x}}{8}-\frac {1}{4} e^{-2 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 (3 x) \, dx &=\text {Subst}\left (\int \frac {1+x^6}{2 x^3} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1+x^6}{x^3} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^3}+x^3\right ) \, dx,x,e^x\right )\\ &=-\frac {1}{4} e^{-2 x}+\frac {e^{4 x}}{8}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{8} e^{-2 x} \left (-2+e^{6 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 26, normalized size = 1.37
method | result | size |
risch | \(\frac {{\mathrm e}^{4 x}}{8}-\frac {{\mathrm e}^{-2 x}}{4}\) | \(14\) |
default | \(\frac {\sinh \left (2 x \right )}{4}+\frac {\sinh \left (4 x \right )}{8}-\frac {\cosh \left (2 x \right )}{4}+\frac {\cosh \left (4 x \right )}{8}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{8} \, e^{\left (4 \, x\right )} - \frac {1}{4} \, e^{\left (-2 \, 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. 38 vs.
\(2 (13) = 26\).
time = 0.36, size = 38, normalized size = 2.00 \begin {gather*} -\frac {\cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - 3 \, \sinh \left (x\right )^{3}}{8 \, {\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 = 0.10, size = 20, normalized size = 1.05 \begin {gather*} \frac {3 e^{x} \sinh {\left (3 x \right )}}{8} - \frac {e^{x} \cosh {\left (3 x \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{8} \, e^{\left (4 \, x\right )} - \frac {1}{4} \, e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 12, normalized size = 0.63 \begin {gather*} \frac {{\mathrm {e}}^{-2\,x}\,\left ({\mathrm {e}}^{6\,x}-2\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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