Optimal. Leaf size=22 \[ \frac {e^{n \cosh (c (a+b x))}}{b c n} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4422, 2225}
\begin {gather*} \frac {e^{n \cosh (c (a+b x))}}{b c n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 4422
Rubi steps
\begin {align*} \int e^{n \cosh (a c+b c x)} \sinh (c (a+b x)) \, dx &=\frac {\text {Subst}\left (\int e^{n x} \, dx,x,\cosh (c (a+b x))\right )}{b c}\\ &=\frac {e^{n \cosh (c (a+b x))}}{b c n}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^{n \cosh (c (a+b x))}}{b c n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 16.45, size = 39, normalized size = 1.77
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {n \left ({\mathrm e}^{c \left (b x +a \right )}+{\mathrm e}^{-c \left (b x +a \right )}\right )}{2}}}{n b c}\) | \(33\) |
default | \(\frac {\frac {\sinh \left (n \cosh \left (c \left (b x +a \right )\right )\right )}{n}+\frac {\cosh \left (n \cosh \left (c \left (b x +a \right )\right )\right )}{n}}{b c}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^{\left (n \cosh \left (b c x + a c\right )\right )}}{b c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 35, normalized size = 1.59 \begin {gather*} \frac {\cosh \left (n \cosh \left (b c x + a c\right )\right ) + \sinh \left (n \cosh \left (b c x + a c\right )\right )}{b c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{n \cosh {\left (a c + b c x \right )}} \sinh {\left (a c + b c x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 38, normalized size = 1.73 \begin {gather*} \frac {e^{\left (\frac {1}{2} \, n e^{\left (b c x + a c\right )} + \frac {1}{2} \, n e^{\left (-b c x - a c\right )}\right )}}{b c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.75, size = 38, normalized size = 1.73 \begin {gather*} \frac {{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}}{b\,c\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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