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 = 0.10, antiderivative size = 46, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 18.90, 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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fricas [A] time = 0.47, size = 70, normalized size = 1.52 \[ \frac {2 \, \left (\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}\right )^{m} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{{\left (2 \, f m + f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \sqrt {-c \sec \left (f x + e\right ) + c} {\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.86, size = 0, normalized size = 0.00 \[ \int \sec \left (f x +e \right ) \left (a +a \sec \left (f x +e \right )\right )^{m} \sqrt {c -c \sec \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 114, normalized size = 2.48 \[ \frac {2^{m + \frac {3}{2}} \left (-a\right )^{m} \sqrt {c} e^{\left (-m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )\right )}}{f {\left (2 \, m + 1\right )} \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1} \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}}{\cos \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{m} \sqrt {- c \left (\sec {\left (e + f x \right )} - 1\right )} \sec {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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