Integrand size = 14, antiderivative size = 75 \[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\frac {3 \sqrt {2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{2},1,\frac {7}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 75, 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} \[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\frac {3 \sqrt {2} \tan (c+d x) \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{2},1,\frac {7}{6},\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{d \sqrt {1-\sec (c+d x)} \sqrt [3]{a \sec (c+d x)+a}} \]
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Rule 141
Rule 3863
Rule 3864
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{1+\sec (c+d x)} \int \frac {1}{\sqrt [3]{1+\sec (c+d x)}} \, dx}{\sqrt [3]{a+a \sec (c+d x)}} \\ & = -\frac {\tan (c+d x) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x (1+x)^{5/6}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt [6]{1+\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}} \\ & = \frac {3 \sqrt {2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{2},1,\frac {7}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(718\) vs. \(2(75)=150\).
Time = 4.78 (sec) , antiderivative size = 718, normalized size of antiderivative = 9.57 \[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\frac {45 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{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 ) \cos (c+d x) (1+\sec (c+d x))^2 \tan \left (\frac {1}{2} (c+d x)\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{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 {1}{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 )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{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 )}{d \sqrt [3]{a (1+\sec (c+d x))} \left (40 \left (3 \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{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 )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{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 {1}{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 {1}{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 ) (1-10 \cos (c+d x)+3 \cos (2 (c+d x)))-5 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{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 ) (1-10 \cos (c+d x)+3 \cos (2 (c+d x)))-24 \left (9 \operatorname {AppellF1}\left (\frac {5}{2},-\frac {1}{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 )+3 \operatorname {AppellF1}\left (\frac {5}{2},\frac {2}{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 )-\operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{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 {1}{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 \sec (c+d x)-3 \sin (c+d x) \tan (c+d x)-\tan ^2(c+d x)\right )\right )} \]
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\[\int \frac {1}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt [3]{a \sec {\left (c + d x \right )} + a}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]
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