Optimal. Leaf size=122 \[ -\frac {64 c^3 \tan (e+f x) (a \sec (e+f x)+a)}{105 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^2 \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{35 f}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f} \]
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Rubi [A] time = 0.20, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3955, 3953} \[ -\frac {64 c^3 \tan (e+f x) (a \sec (e+f x)+a)}{105 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^2 \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{35 f}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{7 f} \]
Antiderivative was successfully verified.
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Rule 3953
Rule 3955
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx &=-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{7 f}+\frac {1}{7} (8 c) \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac {16 c^2 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{35 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{7 f}+\frac {1}{35} \left (32 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \, dx\\ &=-\frac {64 c^3 (a+a \sec (e+f x)) \tan (e+f x)}{105 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^2 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{35 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{7 f}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 76, normalized size = 0.62 \[ \frac {2 a c^2 \cos ^2\left (\frac {1}{2} (e+f x)\right ) (-108 \cos (e+f x)+71 \cos (2 (e+f x))+101) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt {c-c \sec (e+f x)}}{105 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 105, normalized size = 0.86 \[ \frac {2 \, {\left (71 \, a c^{2} \cos \left (f x + e\right )^{4} + 88 \, a c^{2} \cos \left (f x + e\right )^{3} - 22 \, a c^{2} \cos \left (f x + e\right )^{2} - 24 \, a c^{2} \cos \left (f x + e\right ) + 15 \, a c^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{105 \, f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.41, size = 86, normalized size = 0.70 \[ \frac {16 \, \sqrt {2} {\left (35 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} + 42 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3} + 15 \, c^{4}\right )} a c^{2}}{105 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {7}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.18, size = 73, normalized size = 0.60 \[ \frac {2 a \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\sin ^{3}\left (f x +e \right )\right ) \left (71 \left (\cos ^{2}\left (f x +e \right )\right )-54 \cos \left (f x +e \right )+15\right )}{105 f \left (-1+\cos \left (f x +e \right )\right )^{4} \cos \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [B] time = 6.00, size = 384, normalized size = 3.15 \[ \frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,2{}\mathrm {i}}{f}+\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,142{}\mathrm {i}}{105\,f}\right )}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,16{}\mathrm {i}}{7\,f}-\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,16{}\mathrm {i}}{7\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,8{}\mathrm {i}}{5\,f}-\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,184{}\mathrm {i}}{35\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^2\,4{}\mathrm {i}}{3\,f}+\frac {a\,c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,244{}\mathrm {i}}{105\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \]
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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