Optimal. Leaf size=142 \[ \frac{128 c^4 \tan (e+f x)}{5 a f \sqrt{c-c \sec (e+f x)}}+\frac{32 c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{5 a f}+\frac{12 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 a f}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)} \]
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Rubi [A] time = 0.233667, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {3954, 3793, 3792} \[ \frac{128 c^4 \tan (e+f x)}{5 a f \sqrt{c-c \sec (e+f x)}}+\frac{32 c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{5 a f}+\frac{12 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 a f}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{f (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 3954
Rule 3793
Rule 3792
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{a+a \sec (e+f x)} \, dx &=\frac{2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac{(6 c) \int \sec (e+f x) (c-c \sec (e+f x))^{5/2} \, dx}{a}\\ &=\frac{12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac{2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac{\left (48 c^2\right ) \int \sec (e+f x) (c-c \sec (e+f x))^{3/2} \, dx}{5 a}\\ &=\frac{32 c^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{5 a f}+\frac{12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac{2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac{\left (64 c^3\right ) \int \sec (e+f x) \sqrt{c-c \sec (e+f x)} \, dx}{5 a}\\ &=\frac{128 c^4 \tan (e+f x)}{5 a f \sqrt{c-c \sec (e+f x)}}+\frac{32 c^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{5 a f}+\frac{12 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 a f}+\frac{2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{f (a+a \sec (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.756324, size = 86, normalized size = 0.61 \[ -\frac{c^3 (245 \cos (e+f x)+86 \cos (2 (e+f x))+91 \cos (3 (e+f x))+90) \cot \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)}}{10 a f (\cos (e+f x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.187, size = 83, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 182\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}+86\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-14\,\cos \left ( fx+e \right ) +2 \right ) \cos \left ( fx+e \right ) }{5\,fa\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56788, size = 220, normalized size = 1.55 \begin{align*} \frac{8 \,{\left (16 \, \sqrt{2} c^{\frac{7}{2}} - \frac{56 \, \sqrt{2} c^{\frac{7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{70 \, \sqrt{2} c^{\frac{7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{35 \, \sqrt{2} c^{\frac{7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{5 \, \sqrt{2} c^{\frac{7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{5 \, a f{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.481039, size = 209, normalized size = 1.47 \begin{align*} -\frac{2 \,{\left (91 \, c^{3} \cos \left (f x + e\right )^{3} + 43 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{5 \, a f \cos \left (f x + e\right )^{2} \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 = 2.06639, size = 151, normalized size = 1.06 \begin{align*} -\frac{8 \, \sqrt{2}{\left (5 \, \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} c - \frac{15 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} + 5 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} c^{3} + c^{4}}{{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{5}{2}}}\right )} c^{2}}{5 \, a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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