Optimal. Leaf size=83 \[ \frac {\sqrt {2} F_1\left (\frac {1}{2}+m;\frac {1}{2},1;\frac {3}{2}+m;\frac {1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (a+a \sec (e+f x))^m \tan (e+f x)}{f (1+2 m) \sqrt {1-\sec (e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3864, 3863,
141} \begin {gather*} \frac {\sqrt {2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left (m+\frac {1}{2};\frac {1}{2},1;m+\frac {3}{2};\frac {1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right )}{f (2 m+1) \sqrt {1-\sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 3863
Rule 3864
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^m \, dx &=\left ((1+\sec (e+f x))^{-m} (a+a \sec (e+f x))^m\right ) \int (1+\sec (e+f x))^m \, dx\\ &=-\frac {\left ((1+\sec (e+f x))^{-\frac {1}{2}-m} (a+a \sec (e+f x))^m \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m}}{\sqrt {1-x} x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {1-\sec (e+f x)}}\\ &=\frac {\sqrt {2} F_1\left (\frac {1}{2}+m;\frac {1}{2},1;\frac {3}{2}+m;\frac {1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (a+a \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] Leaf count is larger than twice the leaf count of optimal. \(711\) vs. \(2(83)=166\).
time = 6.79, size = 711, normalized size = 8.57 \begin {gather*} \frac {30 F_1\left (\frac {1}{2};m,1;\frac {3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right ) \cos (e+f x) (a (1+\sec (e+f x)))^m \sin (e+f x) \left (3 F_1\left (\frac {1}{2};m,1;\frac {3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left (F_1\left (\frac {3}{2};m,2;\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 F_1\left (\frac {3}{2};1+m,1;\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 ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f \left (45 F_1\left (\frac {1}{2};m,1;\frac {3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ){}^2 \cos ^2\left (\frac {1}{2} (e+f x)\right ) (1+2 m-2 m \cos (e+f x)+\cos (2 (e+f x)))+6 F_1\left (\frac {1}{2};m,1;\frac {3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right ) \left (-5 F_1\left (\frac {3}{2};m,2;\frac {5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+2 m-2 (2+m) \cos (e+f x)+\cos (2 (e+f x)))+5 m F_1\left (\frac {3}{2};1+m,1;\frac {5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+2 m-2 (2+m) \cos (e+f x)+\cos (2 (e+f x)))-48 \left (2 F_1\left (\frac {5}{2};m,3;\frac {7}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 m F_1\left (\frac {5}{2};1+m,2;\frac {7}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+m (1+m) F_1\left (\frac {5}{2};2+m,1;\frac {7}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cot (e+f x) \csc (e+f x) \sin ^4\left (\frac {1}{2} (e+f x)\right )\right )+40 \left (F_1\left (\frac {3}{2};m,2;\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 F_1\left (\frac {3}{2};1+m,1;\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 \cos (e+f x) \sin ^2\left (\frac {1}{2} (e+f x)\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (a +a \sec \left (f x +e \right )\right )^{m}\, dx\]
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 (a \sec {\left (e + f x \right )} + a\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 (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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