Optimal. Leaf size=89 \[ \frac {\sqrt {2} \tan (e+f x) (1-\sec (e+f x))^m F_1\left (m+\frac {1}{2};1-n,\frac {1}{2};m+\frac {3}{2};1-\sec (e+f x),\frac {1}{2} (1-\sec (e+f x))\right )}{f (2 m+1) \sqrt {\sec (e+f x)+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3826, 133} \[ \frac {\sqrt {2} \tan (e+f x) (1-\sec (e+f x))^m F_1\left (m+\frac {1}{2};1-n,\frac {1}{2};m+\frac {3}{2};1-\sec (e+f x),\frac {1}{2} (1-\sec (e+f x))\right )}{f (2 m+1) \sqrt {\sec (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 3826
Rubi steps
\begin {align*} \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx &=\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(1-x)^{-1+n} x^{-\frac {1}{2}+m}}{\sqrt {2-x}} \, dx,x,1-\sec (e+f x)\right )}{f \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}}\\ &=\frac {\sqrt {2} F_1\left (\frac {1}{2}+m;1-n,\frac {1}{2};\frac {3}{2}+m;1-\sec (e+f x),\frac {1}{2} (1-\sec (e+f x))\right ) (1-\sec (e+f x))^m \tan (e+f x)}{f (1+2 m) \sqrt {1+\sec (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 2.34, size = 255, normalized size = 2.87 \[ \frac {(2 m+3) \sin (e+f x) (1-\sec (e+f x))^m \sec ^n(e+f x) F_1\left (m+\frac {1}{2};m+n,1-n;m+\frac {3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (2 m+1) \left (2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left ((n-1) F_1\left (m+\frac {3}{2};m+n,2-n;m+\frac {5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(m+n) F_1\left (m+\frac {3}{2};m+n+1,1-n;m+\frac {5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )+(2 m+3) F_1\left (m+\frac {1}{2};m+n,1-n;m+\frac {3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-\sec \left (f x + e\right ) + 1\right )}^{m} \sec \left (f x + e\right )^{n}, x\right ) \]
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 \[ \int {\left (-\sec \left (f x + e\right ) + 1\right )}^{m} \sec \left (f x + e\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.66, size = 0, normalized size = 0.00 \[ \int \left (1-\sec \left (f x +e \right )\right )^{m} \left (\sec ^{n}\left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (-\sec \left (f x + e\right ) + 1\right )}^{m} \sec \left (f x + e\right )^{n}\,{d x} \]
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 \[ \int {\left (1-\frac {1}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (1 - \sec {\left (e + f x \right )}\right )^{m} \sec ^{n}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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