Optimal. Leaf size=43 \[ \frac {a \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.13, antiderivative size = 43, 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 = {3953} \[ \frac {a \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3953
Rubi steps
\begin {align*} \int \sec (e+f x) \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx &=\frac {a (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 0.47, size = 87, normalized size = 2.02 \[ \frac {c^2 (-6 \cos (e+f x)+3 \cos (2 (e+f x))+5) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt {a (\sec (e+f x)+1)} \sqrt {c-c \sec (e+f x)}}{12 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 93, normalized size = 2.16 \[ \frac {{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \]
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 [B] time = 1.96, size = 82, normalized size = 1.91 \[ -\frac {\sin \left (f x +e \right ) \left (7 \left (\cos ^{2}\left (f x +e \right )\right )-4 \cos \left (f x +e \right )+1\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}}{3 f \left (-1+\cos \left (f x +e \right )\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 638, normalized size = 14.84 \[ \frac {2 \, {\left (30 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 9 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - 3 \, c^{2} \sin \left (f x + e\right ) - {\left (3 \, c^{2} \sin \left (5 \, f x + 5 \, e\right ) - 6 \, c^{2} \sin \left (4 \, f x + 4 \, e\right ) + 10 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) - 6 \, c^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 \, c^{2} \sin \left (f x + e\right )\right )} \cos \left (6 \, f x + 6 \, e\right ) + 9 \, {\left (c^{2} \sin \left (4 \, f x + 4 \, e\right ) + c^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (5 \, f x + 5 \, e\right ) - 3 \, {\left (10 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + {\left (3 \, c^{2} \cos \left (5 \, f x + 5 \, e\right ) - 6 \, c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 10 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) - 6 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, c^{2} \cos \left (f x + e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (3 \, c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + c^{2}\right )} \sin \left (5 \, f x + 5 \, e\right ) + 3 \, {\left (10 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \cos \left (f x + e\right ) + 2 \, c^{2}\right )} \sin \left (4 \, f x + 4 \, e\right ) - 10 \, {\left (3 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + c^{2}\right )} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (3 \, c^{2} \cos \left (f x + e\right ) + 2 \, c^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a} \sqrt {c}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, f x + 4 \, e\right ) + 3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (6 \, f x + 6 \, e\right ) + \cos \left (6 \, f x + 6 \, e\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + 9 \, \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 6 \, {\left (\sin \left (4 \, f x + 4 \, e\right ) + \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) + \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, \sin \left (4 \, f x + 4 \, e\right )^{2} + 18 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 136, normalized size = 3.16 \[ \frac {2\,c^2\,\sqrt {\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (10\,\sin \left (e+f\,x\right )-12\,\sin \left (2\,e+2\,f\,x\right )+13\,\sin \left (3\,e+3\,f\,x\right )-6\,\sin \left (4\,e+4\,f\,x\right )+3\,\sin \left (5\,e+5\,f\,x\right )\right )}{3\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-2\,\cos \left (4\,e+4\,f\,x\right )-\cos \left (6\,e+6\,f\,x\right )+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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