Optimal. Leaf size=205 \[ -\frac {(5-2 n) \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right )}{4 a f (2 n+1) \sqrt {a \sec (e+f x)+a}}+\frac {2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};1-\sec (e+f x)\right )}{a f (2 n+1) \sqrt {a \sec (e+f x)+a}}-\frac {\tan (e+f x) (c-c \sec (e+f x))^n}{2 a f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.17, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3912, 103, 156, 65, 68} \[ -\frac {(5-2 n) \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right )}{4 a f (2 n+1) \sqrt {a \sec (e+f x)+a}}+\frac {2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};1-\sec (e+f x)\right )}{a f (2 n+1) \sqrt {a \sec (e+f x)+a}}-\frac {\tan (e+f x) (c-c \sec (e+f x))^n}{2 a f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 103
Rule 156
Rule 3912
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx &=-\frac {(a c \tan (e+f x)) \operatorname {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{x (a+a x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n} \left (2 a c-\frac {1}{2} a c (1-2 n) x\right )}{x (a+a x)} \, dx,x,\sec (e+f x)\right )}{2 a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {(c \tan (e+f x)) \operatorname {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {(c (5-2 n) \tan (e+f x)) \operatorname {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{a+a x} \, dx,x,\sec (e+f x)\right )}{4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(5-2 n) \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x))\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{4 a f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {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)}{a f (1+2 n) \sqrt {a+a \sec (e+f x)}}-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [F] time = 1.76, size = 0, normalized size = 0.00 \[ \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sec \left (f x + e\right ) + a} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -c \sec \left (f x +e \right )\right )^{n}}{\left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{n}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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