Optimal. Leaf size=14 \[ -\frac{\text{ExpIntegralEi}(n \cos (a+b x))}{b} \]
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Rubi [A] time = 0.0215174, antiderivative size = 14, 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 = {4339, 2178} \[ -\frac{\text{Ei}(n \cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 4339
Rule 2178
Rubi steps
\begin{align*} \int e^{n \cos (a+b x)} \tan (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{e^{n x}}{x} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\text{Ei}(n \cos (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0350774, size = 14, normalized size = 1. \[ -\frac{\text{ExpIntegralEi}(n \cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 16, normalized size = 1.1 \begin{align*}{\frac{{\it Ei} \left ( 1,-n\cos \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 = 1.03419, size = 19, normalized size = 1.36 \begin{align*} -\frac{{\rm Ei}\left (n \cos \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 = 2.08915, size = 31, normalized size = 2.21 \begin{align*} -\frac{{\rm Ei}\left (n \cos \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 \cos{\left (a + b x \right )}} \tan{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (n \cos \left (b x + a\right )\right )} \tan \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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