Optimal. Leaf size=128 \[ -\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}{63 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f} \]
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Rubi [A] time = 0.327305, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3955, 3953} \[ -\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}{63 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{3/2}}{9 f} \]
Antiderivative was successfully verified.
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Rule 3955
Rule 3953
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2} \, dx &=-\frac{2 c (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{9 f}+\frac{1}{9} (8 c) \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac{16 c^2 (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{63 f}-\frac{2 c (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{9 f}+\frac{1}{63} \left (32 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{64 c^3 (a+a \sec (e+f x))^2 \tan (e+f x)}{315 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{63 f}-\frac{2 c (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{9 f}\\ \end{align*}
Mathematica [A] time = 1.19244, size = 78, normalized size = 0.61 \[ \frac{4 a^2 c^2 \cos ^4\left (\frac{1}{2} (e+f x)\right ) (-220 \cos (e+f x)+107 \cos (2 (e+f x))+177) \cot \left (\frac{1}{2} (e+f x)\right ) \sec ^4(e+f x) \sqrt{c-c \sec (e+f x)}}{315 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 75, normalized size = 0.6 \begin{align*} -{\frac{2\,{a}^{2} \left ( 107\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-110\,\cos \left ( fx+e \right ) +35 \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{5}}{315\,f \left ( -1+\cos \left ( fx+e \right ) \right ) ^{5} \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.482101, size = 313, normalized size = 2.45 \begin{align*} \frac{2 \,{\left (107 \, a^{2} c^{2} \cos \left (f x + e\right )^{5} + 211 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 26 \, a^{2} c^{2} \cos \left (f x + e\right )^{3} - 118 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} - 5 \, a^{2} c^{2} \cos \left (f x + e\right ) + 35 \, a^{2} c^{2}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{315 \, f \cos \left (f x + e\right )^{4} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.65187, size = 116, normalized size = 0.91 \begin{align*} -\frac{32 \, \sqrt{2}{\left (63 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{2} c^{4} + 90 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} c^{5} + 35 \, c^{6}\right )} a^{2} c}{315 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{9}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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