Optimal. Leaf size=101 \[ -\frac {2^{\frac {1}{2}+n} c F_1\left (-\frac {3}{2};\frac {1}{2}-n,1;-\frac {1}{2};\frac {1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (1-\sec (e+f x))^{\frac {1}{2}-n} (c-c \sec (e+f x))^{-1+n} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2} \]
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Rubi [A]
time = 0.07, 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 = {3997, 142, 141}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rule 3997
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)) \text {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 ) \text {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 = 1.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.15, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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