Optimal. Leaf size=135 \[ \frac {16 c^3 \tan (e+f x)}{15 f \left (a^3 \sec (e+f x)+a^3\right ) \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f (a \sec (e+f x)+a)^3} \]
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Rubi [A] time = 0.35, antiderivative size = 135, 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, 3953} \[ \frac {16 c^3 \tan (e+f x)}{15 f \left (a^3 \sec (e+f x)+a^3\right ) \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{5 f (a \sec (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3953
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))^3} \, dx &=\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {(4 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^2} \, dx}{5 a}\\ &=-\frac {8 c^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\left (8 c^2\right ) \int \frac {\sec (e+f x) \sqrt {c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx}{15 a^2}\\ &=\frac {16 c^3 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}-\frac {8 c^2 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 74, normalized size = 0.55 \[ -\frac {c^2 \cos (e+f x) (20 \cos (e+f x)+7 \cos (2 (e+f x))+37) \cot \left (\frac {1}{2} (e+f x)\right ) \sqrt {c-c \sec (e+f x)}}{15 a^3 f (\cos (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 104, normalized size = 0.77 \[ -\frac {2 \, {\left (7 \, c^{2} \cos \left (f x + e\right )^{3} + 10 \, c^{2} \cos \left (f x + e\right )^{2} + 15 \, c^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} + 2 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.15, size = 93, normalized size = 0.69 \[ -\frac {\frac {15 \, \sqrt {2} \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}}{a^{3}} + \frac {3 \, \sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} + 10 \, \sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c}{a^{3}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.82, size = 65, normalized size = 0.48 \[ -\frac {2 \left (7 \left (\cos ^{2}\left (f x +e \right )\right )+10 \cos \left (f x +e \right )+15\right ) \left (\cos ^{3}\left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{15 a^{3} f \sin \left (f x +e \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 189, normalized size = 1.40 \[ -\frac {8 \, \sqrt {2} c^{\frac {5}{2}} - \frac {20 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {5 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {3 \, \sqrt {2} c^{\frac {5}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}}}{15 \, a^{3} f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.31, size = 456, normalized size = 3.38 \[ -\frac {c^2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\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}}}\,14{}\mathrm {i}}{15\,a^3\,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 )}+\frac {c^2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\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}}}\,16{}\mathrm {i}}{15\,a^3\,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 )}^2}-\frac {c^2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\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}}}\,112{}\mathrm {i}}{15\,a^3\,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}+\frac {c^2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\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}}}\,64{}\mathrm {i}}{5\,a^3\,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 )}^4}-\frac {c^2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\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}}}\,32{}\mathrm {i}}{5\,a^3\,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 )}^5} \]
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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