Optimal. Leaf size=51 \[ \frac {8 e^{(m+3 i) x} \, _2F_1\left (3,\frac {1}{2} (3-i m);\frac {1}{2} (5-i m);-e^{2 i x}\right )}{m+3 i} \]
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Rubi [A] time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.51, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4448, 4451} \[ (-m+i) \left (-e^{(m+i) x}\right ) \text {Hypergeometric2F1}\left (1,\frac {1}{2} (1-i m),\frac {1}{2} (3-i m),-e^{2 i x}\right )-\frac {1}{2} m e^{m x} \sec (x)+\frac {1}{2} e^{m x} \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Rule 4448
Rule 4451
Rubi steps
\begin {align*} \int e^{m x} \sec ^3(x) \, dx &=-\frac {1}{2} e^{m x} m \sec (x)+\frac {1}{2} e^{m x} \sec (x) \tan (x)+\frac {1}{2} \left (1+m^2\right ) \int e^{m x} \sec (x) \, dx\\ &=-e^{(i+m) x} (i-m) \, _2F_1\left (1,\frac {1}{2} (1-i m);\frac {1}{2} (3-i m);-e^{2 i x}\right )-\frac {1}{2} e^{m x} m \sec (x)+\frac {1}{2} e^{m x} \sec (x) \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 66, normalized size = 1.29 \[ \frac {1}{2} e^{m x} \left (\sec (x) (\tan (x)-m)+2 (m-i) e^{i x} \, _2F_1\left (1,\frac {1}{2}-\frac {i m}{2};\frac {3}{2}-\frac {i m}{2};-e^{2 i x}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \sec ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (m x\right )}}{\cos \relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {e^{\left (m x\right )}}{\cos \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{m x}}{\cos \relax (x )^{3}}\, dx\]
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 \[ \int \frac {{\mathrm {e}}^{m\,x}}{{\cos \relax (x)}^3} \,d x \]
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 \[ \int \frac {e^{m x}}{\cos ^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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