Optimal. Leaf size=92 \[ \frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2}+n;\frac {1}{2}-m,1;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {1+\sec (e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3997, 141}
\begin {gather*} \frac {2^{m+\frac {1}{2}} \tan (e+f x) (c-c \sec (e+f x))^n F_1\left (n+\frac {1}{2};\frac {1}{2}-m,1;n+\frac {3}{2};\frac {1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right )}{f (2 n+1) \sqrt {\sec (e+f x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 3997
Rubi steps
\begin {align*} \int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, dx &=-\frac {(c \tan (e+f x)) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (c-c x)^{-\frac {1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {1+\sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=\frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2}+n;\frac {1}{2}-m,1;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {1+\sec (e+f x)}}\\ \end {align*}
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Mathematica [F]
time = 1.10, size = 0, normalized size = 0.00 \begin {gather*} \int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, 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 \[\int \left (1+\sec \left (f x +e \right )\right )^{m} \left (c -c \sec \left (f x +e \right )\right )^{n}\, dx\]
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 \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 \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{n} \left (\sec {\left (e + f x \right )} + 1\right )^{m}\, 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 {\left (\frac {1}{\cos \left (e+f\,x\right )}+1\right )}^m\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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