Optimal. Leaf size=139 \[ -\frac {\, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x))\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3997, 88, 67,
70} \begin {gather*} \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 )}{f (2 n+1) \sqrt {a \sec (e+f x)+a}}-\frac {\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 )}{f (2 n+1) \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 88
Rule 3997
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^n}{\sqrt {a+a \sec (e+f x)}} \, dx &=-\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{x (a+a x)} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(c \tan (e+f x)) \text {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 {(a c \tan (e+f x)) \text {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{a+a x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {\, _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)}{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)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [F]
time = 1.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c-c \sec (e+f x))^n}{\sqrt {a+a \sec (e+f x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (c -c \sec \left (f x +e \right )\right )^{n}}{\sqrt {a +a \sec \left (f x +e \right )}}\, dx\]
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*} \int \frac {\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{n}}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}}\, dx \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}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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