Optimal. Leaf size=19 \[ -\frac{\text{ExpIntegralEi}(n \cos (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.0219697, antiderivative size = 19, normalized size of antiderivative = 1., 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 = {4339, 2178} \[ -\frac{\text{Ei}(n \cos (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 4339
Rule 2178
Rubi steps
\begin{align*} \int e^{n \cos (a c+b c x)} \tan (c (a+b x)) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{e^{n x}}{x} \, dx,x,\cos (c (a+b x))\right )}{b c}\\ &=-\frac{\text{Ei}(n \cos (c (a+b x)))}{b c}\\ \end{align*}
Mathematica [A] time = 0.0607347, size = 19, normalized size = 1. \[ -\frac{\text{ExpIntegralEi}(n \cos (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 22, normalized size = 1.2 \begin{align*}{\frac{{\it Ei} \left ( 1,-n\cos \left ( bcx+ac \right ) \right ) }{cb}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08073, size = 27, normalized size = 1.42 \begin{align*} -\frac{{\rm Ei}\left (n \cos \left (b c x + a c\right )\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3288, size = 42, normalized size = 2.21 \begin{align*} -\frac{{\rm Ei}\left (n \cos \left (b c x + a c\right )\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 c x + a c\right )\right )} \tan \left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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