Optimal. Leaf size=13 \[ \frac{\text{ExpIntegralEi}(n \sin (a+b x))}{b} \]
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Rubi [A] time = 0.0202975, antiderivative size = 13, normalized size of antiderivative = 1., 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 = {4338, 2178} \[ \frac{\text{Ei}(n \sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 4338
Rule 2178
Rubi steps
\begin{align*} \int e^{n \sin (a+b x)} \cot (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^{n x}}{x} \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\text{Ei}(n \sin (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0424549, size = 13, normalized size = 1. \[ \frac{\text{ExpIntegralEi}(n \sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 17, normalized size = 1.3 \begin{align*} -{\frac{{\it Ei} \left ( 1,-n\sin \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07049, size = 18, normalized size = 1.38 \begin{align*} \frac{{\rm Ei}\left (n \sin \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16819, size = 30, normalized size = 2.31 \begin{align*} \frac{{\rm Ei}\left (n \sin \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \sin{\left (a + b x \right )}} \cot{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08886, size = 18, normalized size = 1.38 \begin{align*} \frac{{\rm Ei}\left (n \sin \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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