Optimal. Leaf size=123 \[ -\frac {16 c^3 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.27, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 = {3954, 3792} \[ -\frac {16 c^3 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3954
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{(a+a \sec (e+f x))^2} \, dx &=\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {(4 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx}{3 a}\\ &=-\frac {8 c^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (8 c^2\right ) \int \sec (e+f x) \sqrt {c-c \sec (e+f x)} \, dx}{3 a^2}\\ &=-\frac {16 c^3 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 68, normalized size = 0.55 \[ \frac {c^2 (36 \cos (e+f x)+11 \cos (2 (e+f x))+17) \cot \left (\frac {1}{2} (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}{3 a^2 f (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 82, normalized size = 0.67 \[ \frac {2 \, {\left (11 \, c^{2} \cos \left (f x + e\right )^{2} + 18 \, c^{2} \cos \left (f x + e\right ) + 3 \, c^{2}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.77, size = 102, normalized size = 0.83 \[ -\frac {2 \, \sqrt {2} c^{2} {\left (\frac {3 \, c}{\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a^{2}} - \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} a^{4} c^{2} + 6 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a^{4} c^{3}}{a^{6} c^{3}}\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.67, size = 75, normalized size = 0.61 \[ -\frac {2 \left (11 \left (\cos ^{2}\left (f x +e \right )\right )+18 \cos \left (f x +e \right )+3\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{3 a^{2} f \sin \left (f x +e \right )^{3} \left (-1+\cos \left (f x +e \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 163, normalized size = 1.33 \[ \frac {2 \, {\left (8 \, \sqrt {2} c^{\frac {5}{2}} - \frac {20 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {2 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {\sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{3 \, a^{2} f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.54, size = 136, normalized size = 1.11 \[ \frac {2\,c^2\,\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,36{}\mathrm {i}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,34{}\mathrm {i}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,36{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,11{}\mathrm {i}+11{}\mathrm {i}\right )}{3\,a^2\,f\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}^3} \]
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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