Optimal. Leaf size=78 \[ \frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {i e^{c (a+b x)}}{b c} \]
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Rubi [A] time = 0.08, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4442, 2194, 2251} \[ \frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}-\frac {i e^{c (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 4442
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tan (d+e x) \, dx &=i \int \left (-e^{c (a+b x)}+\frac {2 e^{c (a+b x)}}{1+e^{2 i (d+e x)}}\right ) \, dx\\ &=-\left (i \int e^{c (a+b x)} \, dx\right )+2 i \int \frac {e^{c (a+b x)}}{1+e^{2 i (d+e x)}} \, dx\\ &=-\frac {i e^{c (a+b x)}}{b c}+\frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c}\\ \end {align*}
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Mathematica [B] time = 0.47, size = 166, normalized size = 2.13 \[ \frac {e^{c (a+b x)} \left (2 b c e^{2 i (d+e x)} \, _2F_1\left (1,1-\frac {i b c}{2 e};2-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )-(b c+2 i e) \left (2 e^{2 i d} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )-e^{2 i d}+1\right )\right )}{b c \left (1+e^{2 i d}\right ) (-2 e+i b c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b c x + a c\right )} \tan \left (e x + d\right ), x\right ) \]
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 \[ \int e^{\left ({\left (b x + a\right )} c\right )} \tan \left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \tan \left (e x +d \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left ({\left (b x + a\right )} c\right )} \tan \left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\mathrm {tan}\left (d+e\,x\right ) \,d x \]
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 \[ e^{a c} \int e^{b c x} \tan {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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