Optimal. Leaf size=101 \[ -\frac {c 2^{n+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac {1}{2}-n} F_1\left (-\frac {3}{2};\frac {1}{2}-n,1;-\frac {1}{2};\frac {1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right ) (c-c \sec (e+f x))^{n-1}}{3 f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3912, 137, 136} \[ -\frac {c 2^{n+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac {1}{2}-n} F_1\left (-\frac {3}{2};\frac {1}{2}-n,1;-\frac {1}{2};\frac {1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right ) (c-c \sec (e+f x))^{n-1}}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rule 3912
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^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)^{5/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {\left (2^{-\frac {1}{2}+n} a c (c-c \sec (e+f x))^{-1+n} \left (\frac {c-c \sec (e+f x)}{c}\right )^{\frac {1}{2}-n} \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}-\frac {x}{2}\right )^{-\frac {1}{2}+n}}{x (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {2^{\frac {1}{2}+n} c F_1\left (-\frac {3}{2};\frac {1}{2}-n,1;-\frac {1}{2};\frac {1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (1-\sec (e+f x))^{\frac {1}{2}-n} (c-c \sec (e+f x))^{-1+n} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}\\ \end {align*}
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Mathematica [F] time = 1.73, size = 0, normalized size = 0.00 \[ \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.75, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -c \sec \left (f x +e \right )\right )^{n}}{\left (a +a \sec \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2} \,d x \]
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 \[ \frac {\int \frac {\left (- c \sec {\left (e + f x \right )} + c\right )^{n}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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