Optimal. Leaf size=171 \[ -\frac{256 c^4 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 f \sqrt{c-c \sec (e+f x)}}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{429 f}-\frac{24 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{143 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}{13 f} \]
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Rubi [A] time = 0.443859, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3955, 3953} \[ -\frac{256 c^4 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 f \sqrt{c-c \sec (e+f x)}}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{429 f}-\frac{24 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{143 f}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}{13 f} \]
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))^3 (c-c \sec (e+f x))^{7/2} \, dx &=-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}+\frac{1}{13} (12 c) \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx\\ &=-\frac{24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}+\frac{1}{143} \left (96 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac{64 c^3 (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{429 f}-\frac{24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}+\frac{1}{429} \left (128 c^3\right ) \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{256 c^4 (a+a \sec (e+f x))^3 \tan (e+f x)}{3003 f \sqrt{c-c \sec (e+f x)}}-\frac{64 c^3 (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{429 f}-\frac{24 c^2 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{143 f}-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{13 f}\\ \end{align*}
Mathematica [A] time = 2.59016, size = 88, normalized size = 0.51 \[ \frac{4 a^3 c^3 \cos ^6\left (\frac{1}{2} (e+f x)\right ) (6285 \cos (e+f x)-2842 \cos (2 (e+f x))+835 \cos (3 (e+f x))-3766) \cot \left (\frac{1}{2} (e+f x)\right ) \sec ^6(e+f x) \sqrt{c-c \sec (e+f x)}}{3003 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.218, size = 85, normalized size = 0.5 \begin{align*}{\frac{2\,{a}^{3} \left ( 835\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-1421\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+945\,\cos \left ( fx+e \right ) -231 \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{7}}{3003\,f \left ( -1+\cos \left ( fx+e \right ) \right ) ^{7} \left ( \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 [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 = 0.504792, size = 400, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (835 \, a^{3} c^{3} \cos \left (f x + e\right )^{7} + 1919 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 271 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} - 1637 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} - 103 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 973 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 21 \, a^{3} c^{3} \cos \left (f x + e\right ) - 231 \, a^{3} c^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3003 \, f \cos \left (f x + e\right )^{6} \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 = 5.74682, size = 153, normalized size = 0.89 \begin{align*} \frac{128 \, \sqrt{2}{\left (429 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{3} c^{5} + 1001 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{2} c^{6} + 819 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} c^{7} + 231 \, c^{8}\right )} a^{3} c^{2}}{3003 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{13}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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