Optimal. Leaf size=39 \[ \frac {2 c \tan (e+f x)}{f (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.07, 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 = {4038}
\begin {gather*} \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4038
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 \begin {gather*} -\frac {2 \cot (e+f x) \sqrt {c-c \sec (e+f x)}}{a f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.55, size = 43, normalized size = 1.10
method | result | size |
default | \(-\frac {2 \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right )}{a f \sin \left (f x +e \right )}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (40) = 80\).
time = 0.53, size = 90, normalized size = 2.31 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.39, size = 49, normalized size = 1.26 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 59, normalized size = 1.51 \begin {gather*} -\frac {\sqrt {2} \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a f} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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