Optimal. Leaf size=78 \[ -\frac {i e^{c (a+b x)}}{b c}+\frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};-e^{2 i (d+e x)}\right )}{b c} \]
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Rubi [A]
time = 0.05, 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 = {4527, 2225,
2283} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 4527
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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(166\) vs. \(2(78)=156\).
time = 0.46, size = 166, normalized size = 2.13 \begin {gather*} \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 (1-e^{2 i d}+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 )\right )\right )}{b c (i b c-2 e) \left (1+e^{2 i d}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} e^{a c} \int e^{b c x} \tan {\left (d + e x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\mathrm {tan}\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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