Optimal. Leaf size=46 \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.101803, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3953} \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3953
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^m \sqrt{c-c \sec (e+f x)} \, dx &=-\frac{2 c (a+a \sec (e+f x))^m \tan (e+f x)}{f (1+2 m) \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 18.7722, size = 163, normalized size = 3.54 \[ \frac{\sqrt{2} e^{-\frac{1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right ) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \left (\frac{\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}\right )^m \csc \left (\frac{1}{2} (e+f x)\right ) \sqrt{c-c \sec (e+f x)} (\sec (e+f x)+1)^{-m} (a (\sec (e+f x)+1))^m}{(2 f m+f) \sqrt{\sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.313, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( fx+e \right ) \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m}\sqrt{c-c\sec \left ( fx+e \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55983, size = 154, normalized size = 3.35 \begin{align*} \frac{2^{m + \frac{3}{2}} \left (-a\right )^{m} \sqrt{c} e^{\left (-m \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )\right )}}{f{\left (2 \, m + 1\right )} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.474301, size = 171, normalized size = 3.72 \begin{align*} \frac{2 \, \left (\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}\right )^{m} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}{\left (\cos \left (f x + e\right ) + 1\right )}}{{\left (2 \, f m + f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-c \sec \left (f x + e\right ) + c}{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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