Optimal. Leaf size=17 \[ \frac {(b \sec (e+f x))^m}{f m} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 32
Rule 2606
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*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ \frac {(b \sec (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 19, normalized size = 1.12 \[ \frac {\left (\frac {b}{\cos \left (f x + e\right )}\right )^{m}}{f m} \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 18, normalized size = 1.06 \[ \frac {\left (b \sec \left (f x +e \right )\right )^{m}}{f m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 20, normalized size = 1.18 \[ \frac {b^{m} \cos \left (f x + e\right )^{-m}}{f m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 19, normalized size = 1.12 \[ \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m}{f\,m} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 44, normalized size = 2.59 \[ \begin {cases} x \tan {\relax (e )} & \text {for}\: f = 0 \wedge m = 0 \\x \left (b \sec {\relax (e )}\right )^{m} \tan {\relax (e )} & \text {for}\: f = 0 \\\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} & \text {for}\: m = 0 \\\frac {b^{m} \sec ^{m}{\left (e + f x \right )}}{f m} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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