Optimal. Leaf size=81 \[ -\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)}{15 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{5 f} \]
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Rubi [A] time = 0.13, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3955, 3953} \[ -\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)}{15 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{5 f} \]
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))^{3/2} \, dx &=-\frac {2 c (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{5 f}+\frac {1}{5} (4 c) \int \sec (e+f x) (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \, dx\\ &=-\frac {8 c^2 (a+a \sec (e+f x)) \tan (e+f x)}{15 f \sqrt {c-c \sec (e+f x)}}-\frac {2 c (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 64, normalized size = 0.79 \[ \frac {4 a c \cos ^2\left (\frac {1}{2} (e+f x)\right ) (7 \cos (e+f x)-3) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt {c-c \sec (e+f x)}}{15 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 82, normalized size = 1.01 \[ \frac {2 \, {\left (7 \, a c \cos \left (f x + e\right )^{3} + 11 \, a c \cos \left (f x + e\right )^{2} + a c \cos \left (f x + e\right ) - 3 \, a c\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{15 \, f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.54, size = 58, normalized size = 0.72 \[ \frac {8 \, \sqrt {2} {\left (5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3} + 3 \, c^{4}\right )} a}{15 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.21, size = 63, normalized size = 0.78 \[ \frac {2 a \left (7 \cos \left (f x +e \right )-3\right ) \left (\sin ^{3}\left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{15 f \left (-1+\cos \left (f x +e \right )\right )^{3} \cos \left (f x +e \right )} \]
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 = 5.36, size = 120, normalized size = 1.48 \[ -\frac {2\,a\,c\,{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^3\,\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (7+7\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-6\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\right )}{15\,f\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )}\, dx + \int \left (- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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