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 = 0.0363634, antiderivative size = 77, normalized size of antiderivative = 1.51, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}
Mathematica [A] time = 0.0470558, 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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Maple [F] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{mx}}}{ \left ( \cos \left ( x \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (m x\right )}}{\cos \left (x\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{m x}}{\cos ^{3}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (m x\right )}}{\cos \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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