Optimal. Leaf size=13 \[ \frac {\text {Ei}(n \sinh (a+b x))}{b} \]
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Rubi [A] time = 0.02, 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 = {4340, 2178} \[ \frac {\text {Ei}(n \sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 4340
Rubi steps
\begin {align*} \int e^{n \sinh (a+b x)} \coth (a+b x) \, dx &=\frac {\operatorname {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 = 0.04, size = 13, normalized size = 1.00 \[ \frac {\text {Ei}(n \sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 13, normalized size = 1.00 \[ \frac {{\rm Ei}\left (n \sinh \left (b x + a\right )\right )}{b} \]
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 \[ \int \coth \left (b x + a\right ) e^{\left (n \sinh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 17, normalized size = 1.31 \[ -\frac {\Ei \left (1, -n \sinh \left (b x +a \right )\right )}{b} \]
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 \[ \int \coth \left (b x + a\right ) e^{\left (n \sinh \left (b x + a\right )\right )}\,{d x} \]
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 \[ \int \mathrm {coth}\left (a+b\,x\right )\,{\mathrm {e}}^{n\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int e^{n \sinh {\left (a + b x \right )}} \coth {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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