Optimal. Leaf size=14 \[ -\frac {\text {Ei}(n \cos (a+b x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 14, 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 = {4339, 2178} \[ -\frac {\text {Ei}(n \cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 4339
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*}
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Mathematica [A] time = 0.04, size = 14, normalized size = 1.00 \[ -\frac {\text {Ei}(n \cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.49, size = 14, normalized size = 1.00 \[ -\frac {{\rm Ei}\left (n \cos \left (b x + a\right )\right )}{b} \]
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 \[ \int e^{\left (n \cos \left (b x + a\right )\right )} \tan \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 16, normalized size = 1.14 \[ \frac {\Ei \left (1, -n \cos \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 14, normalized size = 1.00 \[ -\frac {{\rm Ei}\left (n \cos \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int {\mathrm {e}}^{n\,\cos \left (a+b\,x\right )}\,\mathrm {tan}\left (a+b\,x\right ) \,d x \]
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 \[ \int e^{n \cos {\left (a + b x \right )}} \tan {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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