Optimal. Leaf size=58 \[ -\frac {e^{m x}}{m}+\frac {4 e^{(m+2 i) x} \, _2F_1\left (2,1-\frac {i m}{2};2-\frac {i m}{2};-e^{2 i x}\right )}{m+2 i} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.47, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4442, 2194, 2251} \[ \frac {4 e^{m x} \text {Hypergeometric2F1}\left (1,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )}{m}-\frac {4 e^{m x} \text {Hypergeometric2F1}\left (2,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )}{m}-\frac {e^{m x}}{m} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 4442
Rubi steps
\begin {align*} \int e^{m x} \tan ^2(x) \, dx &=-\int \left (e^{m x}+\frac {4 e^{m x}}{\left (1+e^{2 i x}\right )^2}-\frac {4 e^{m x}}{1+e^{2 i x}}\right ) \, dx\\ &=-\left (4 \int \frac {e^{m x}}{\left (1+e^{2 i x}\right )^2} \, dx\right )+4 \int \frac {e^{m x}}{1+e^{2 i x}} \, dx-\int e^{m x} \, dx\\ &=-\frac {e^{m x}}{m}+\frac {4 e^{m x} \, _2F_1\left (1,-\frac {i m}{2};1-\frac {i m}{2};-e^{2 i x}\right )}{m}-\frac {4 e^{m x} \, _2F_1\left (2,-\frac {i m}{2};1-\frac {i m}{2};-e^{2 i x}\right )}{m}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 97, normalized size = 1.67 \[ \frac {e^{m x} \left (\frac {i m^2 e^{2 i x} \, _2F_1\left (1,1-\frac {i m}{2};2-\frac {i m}{2};-e^{2 i x}\right )}{m+2 i}-i m \, _2F_1\left (1,-\frac {i m}{2};1-\frac {i m}{2};-e^{2 i x}\right )+m \tan (x)-1\right )}{m} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \tan ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (m x\right )} \tan \relax (x)^{2}, x\right ) \]
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 \[ \int e^{\left (m x\right )} \tan \relax (x)^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{m x} \left (\tan ^{2}\relax (x )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{m\,x}\,{\mathrm {tan}\relax (x)}^2 \,d x \]
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 \[ \int e^{m x} \tan ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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