Optimal. Leaf size=42 \[ -\frac {\tan (e+f x) \sqrt {a \sec (e+f x)+a}}{2 f (c-c \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {3950} \[ -\frac {\tan (e+f x) \sqrt {a \sec (e+f x)+a}}{2 f (c-c \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3950
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx &=-\frac {\sqrt {a+a \sec (e+f x)} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 62, normalized size = 1.48 \[ \frac {\tan \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {a (\sec (e+f x)+1)}}{c f (\sec (e+f x)-1) \sqrt {c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 79, normalized size = 1.88 \[ \frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \sin \left (f x + e\right )} \]
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 = 2.05, size = 60, normalized size = 1.43 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )}{2 f \cos \left (f x +e \right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 514, normalized size = 12.24 \[ -\frac {2 \, {\left ({\left (\sin \left (3 \, f x + 3 \, e\right ) + \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - {\left (\cos \left (3 \, f x + 3 \, e\right ) + \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - 2 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + \sin \left (f x + e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (c^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c^{2} \cos \left (3 \, f x + 3 \, e\right )^{2} + 4 \, c^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, c^{2} \cos \left (f x + e\right )^{2} + c^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c^{2} \sin \left (3 \, f x + 3 \, e\right )^{2} + 4 \, c^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 8 \, c^{2} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 4 \, c^{2} \sin \left (f x + e\right )^{2} - 4 \, c^{2} \cos \left (f x + e\right ) + c^{2} - 2 \, {\left (2 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) - 2 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2 \, c^{2} \cos \left (f x + e\right ) - c^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 \, {\left (2 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) - 2 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 4 \, {\left (2 \, c^{2} \cos \left (f x + e\right ) - c^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (c^{2} \sin \left (3 \, f x + 3 \, e\right ) - c^{2} \sin \left (2 \, f x + 2 \, e\right ) + c^{2} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 8 \, {\left (c^{2} \sin \left (2 \, f x + 2 \, e\right ) - c^{2} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.00, size = 118, normalized size = 2.81 \[ -\frac {2\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (\sin \left (e+f\,x\right )-2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\right )}{c^2\,f\,\left (4\,\cos \left (e+f\,x\right )+4\,\cos \left (2\,e+2\,f\,x\right )-4\,\cos \left (3\,e+3\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )-5\right )} \]
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 \[ \int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sec {\left (e + f x \right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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