Optimal. Leaf size=13 \[ \frac {\text {Ei}(n \sinh (a+b x))}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4425, 2209}
\begin {gather*} \frac {\text {Ei}(n \sinh (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 4425
Rubi steps
\begin {align*} \int e^{n \sinh (a+b x)} \coth (a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {e^{n x}}{x} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac {\text {Ei}(n \sinh (a+b x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} \frac {\text {Ei}(n \sinh (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.00, size = 17, normalized size = 1.31
method | result | size |
derivativedivides | \(-\frac {\expIntegral \left (1, -n \sinh \left (b x +a \right )\right )}{b}\) | \(17\) |
default | \(-\frac {\expIntegral \left (1, -n \sinh \left (b x +a \right )\right )}{b}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 13, normalized size = 1.00 \begin {gather*} \frac {{\rm Ei}\left (n \sinh \left (b x + a\right )\right )}{b} \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 \sinh {\left (a + b x \right )}} \coth {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \mathrm {coth}\left (a+b\,x\right )\,{\mathrm {e}}^{n\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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