Integrand size = 14, antiderivative size = 77 \[ \int (a+a \sec (c+d x))^{2/3} \, dx=\frac {3 \sqrt {2} \operatorname {AppellF1}\left (\frac {7}{6},\frac {1}{2},1,\frac {13}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{7 d \sqrt {1-\sec (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3864, 3863, 141} \[ \int (a+a \sec (c+d x))^{2/3} \, dx=\frac {3 \sqrt {2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} \operatorname {AppellF1}\left (\frac {7}{6},\frac {1}{2},1,\frac {13}{6},\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{7 d \sqrt {1-\sec (c+d x)}} \]
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Rule 141
Rule 3863
Rule 3864
Rubi steps \begin{align*} \text {integral}& = \frac {(a+a \sec (c+d x))^{2/3} \int (1+\sec (c+d x))^{2/3} \, dx}{(1+\sec (c+d x))^{2/3}} \\ & = -\frac {\left ((a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt {1-x} x} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}} \\ & = \frac {3 \sqrt {2} \operatorname {AppellF1}\left (\frac {7}{6},\frac {1}{2},1,\frac {13}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{7 d \sqrt {1-\sec (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(691\) vs. \(2(77)=154\).
Time = 5.21 (sec) , antiderivative size = 691, normalized size of antiderivative = 8.97 \[ \int (a+a \sec (c+d x))^{2/3} \, dx=\frac {45 \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sec (c+d x)))^{5/3} \sin (c+d x) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+2 \left (-3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{3},1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{a d \left (40 \left (3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{3},1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )^2 \sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (15 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (-7+16 \cos (c+d x)-3 \cos (2 (c+d x)))+10 \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{3},1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (7-16 \cos (c+d x)+3 \cos (2 (c+d x)))-24 \left (9 \operatorname {AppellF1}\left (\frac {5}{2},\frac {2}{3},3,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-6 \operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{3},2,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+5 \operatorname {AppellF1}\left (\frac {5}{2},\frac {8}{3},1,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \cos (c+d x) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+135 \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2 \left (3+3 \cos (c+d x)+2 \tan ^2(c+d x)\right )\right )} \]
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\[\int \left (a +a \sec \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
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Timed out. \[ \int (a+a \sec (c+d x))^{2/3} \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x))^{2/3} \, dx=\int \left (a \sec {\left (c + d x \right )} + a\right )^{\frac {2}{3}}\, dx \]
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\[ \int (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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\[ \int (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^{2/3} \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{2/3} \,d x \]
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