Optimal. Leaf size=21 \[ -\frac{\sin (e+f x) \sec ^{m+1}(e+f x)}{f} \]
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Rubi [A] time = 0.0294609, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {4043} \[ -\frac{\sin (e+f x) \sec ^{m+1}(e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin{align*} \int \sec ^m(e+f x) \left (m-(1+m) \sec ^2(e+f x)\right ) \, dx &=-\frac{\sec ^{1+m}(e+f x) \sin (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.253448, size = 21, normalized size = 1. \[ -\frac{\sin (e+f x) \sec ^{m+1}(e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.398, size = 506, normalized size = 24.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.97325, size = 382, normalized size = 18.19 \begin{align*} \frac{2^{m} \cos \left (-{\left (f x + e\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2^{m} \cos \left (-{\left (f x + e\right )} m + m \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) +{\left (2^{m} \cos \left (2 \, f x + 2 \, e\right ) + 2^{m}\right )} \sin \left (-{\left (f x + e\right )}{\left (m + 2\right )} + m \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) -{\left (2^{m} \cos \left (2 \, f x + 2 \, e\right ) + 2^{m}\right )} \sin \left (-{\left (f x + e\right )} m + m \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )}{{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac{1}{2} \, m} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.49293, size = 69, normalized size = 3.29 \begin{align*} -\frac{\frac{1}{\cos \left (f x + e\right )}^{m} \sin \left (f x + e\right )}{f \cos \left (f x + e\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 - m \sec ^{m}{\left (e + f x \right )}\, dx - \int \sec ^{2}{\left (e + f x \right )} \sec ^{m}{\left (e + f x \right )}\, dx - \int m \sec ^{2}{\left (e + f x \right )} \sec ^{m}{\left (e + f 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 -{\left ({\left (m + 1\right )} \sec \left (f x + e\right )^{2} - m\right )} \sec \left (f x + e\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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