Optimal. Leaf size=39 \[ \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 39, 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} \[ \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3953
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) \sqrt {c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx &=\frac {2 c \tan (e+f x)}{f (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 29, normalized size = 0.74 \[ -\frac {2 \cot (e+f x) \sqrt {c-c \sec (e+f x)}}{a f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 45, normalized size = 1.15 \[ -\frac {2 \, \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.12, size = 43, normalized size = 1.10 \[ -\frac {2 \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right )}{a f \sin \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.75, size = 84, normalized size = 2.15 \[ -\frac {\sqrt {2} \sqrt {c} - \frac {\sqrt {2} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a f \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1} \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.77, size = 40, normalized size = 1.03 \[ -\frac {\sin \left (2\,e+2\,f\,x\right )\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}}{a\,f\,{\sin \left (e+f\,x\right )}^2} \]
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 \[ \frac {\int \frac {\sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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