Optimal. Leaf size=119 \[ \frac {2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};1-\sec (e+f x)\right )}{f (2 n+1) \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt {a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.16, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3909, 3912, 65} \[ \frac {2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};1-\sec (e+f x)\right )}{f (2 n+1) \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \tan (e+f x) (c-c \sec (e+f x))^n}{f (2 n+1) \sqrt {a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 3909
Rule 3912
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^n \, dx &=\frac {2 a^2 (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+a \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^n \, dx\\ &=\frac {2 a^2 (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^2 c \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=\frac {2 a^2 (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [F] time = 11.83, size = 0, normalized size = 0.00 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.73, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \left (c -c \sec \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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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