Optimal. Leaf size=19 \[ -\frac {\text {Ei}(n \cos (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 2178
Rule 4339
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*}
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Mathematica [A] time = 0.06, size = 19, normalized size = 1.00 \[ -\frac {\text {Ei}(n \cos (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 20, normalized size = 1.05 \[ -\frac {{\rm Ei}\left (n \cos \left (b c x + a c\right )\right )}{b c} \]
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 \[ \int e^{\left (n \cos \left (b c x + a c\right )\right )} \tan \left ({\left (b x + a\right )} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 22, normalized size = 1.16 \[ \frac {\Ei \left (1, -n \cos \left (b c x +a c \right )\right )}{c b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 20, normalized size = 1.05 \[ -\frac {{\rm Ei}\left (n \cos \left (b c x + a c\right )\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \mathrm {tan}\left (c\,\left (a+b\,x\right )\right )\,{\mathrm {e}}^{n\,\cos \left (a\,c+b\,c\,x\right )} \,d x \]
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 \[ \int e^{n \cos {\left (a c + b c x \right )}} \tan {\left (a c + b c x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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