Optimal. Leaf size=17 \[ \frac{(b \sec (e+f x))^m}{f m} \]
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Rubi [A] time = 0.0210696, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2606, 32} \[ \frac{(b \sec (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 32
Rubi steps
\begin{align*} \int (b \sec (e+f x))^m \tan (e+f x) \, dx &=\frac{b \operatorname{Subst}\left (\int (b x)^{-1+m} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{(b \sec (e+f x))^m}{f m}\\ \end{align*}
Mathematica [A] time = 0.0216875, size = 17, normalized size = 1. \[ \frac{(b \sec (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ( b\sec \left ( fx+e \right ) \right ) ^{m}}{fm}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999073, size = 27, normalized size = 1.59 \begin{align*} \frac{b^{m} \cos \left (f x + e\right )^{-m}}{f m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6066, size = 35, normalized size = 2.06 \begin{align*} \frac{\left (\frac{b}{\cos \left (f x + e\right )}\right )^{m}}{f m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.538158, size = 44, normalized size = 2.59 \begin{align*} \begin{cases} x \tan{\left (e \right )} & \text{for}\: f = 0 \wedge m = 0 \\x \left (b \sec{\left (e \right )}\right )^{m} \tan{\left (e \right )} & \text{for}\: f = 0 \\\frac{\log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} & \text{for}\: m = 0 \\\frac{b^{m} \sec ^{m}{\left (e + f x \right )}}{f m} & \text{otherwise} \end{cases} \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 \left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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