Integrand size = 27, antiderivative size = 76 \[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\sec (c+d x),-\sec (c+d x)\right ) \sqrt [3]{e \sec (c+d x)} \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3913, 3912, 129, 440} \[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {3 \tan (c+d x) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\sec (c+d x),-\sec (c+d x)\right ) \sqrt [3]{e \sec (c+d x)}}{d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \]
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Rule 129
Rule 440
Rule 3912
Rule 3913
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\sec (c+d x)} \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx}{\sqrt {a+a \sec (c+d x)}} \\ & = -\frac {(e \tan (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} (e x)^{2/3} (1+x)} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = -\frac {(3 \tan (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^3}{e}} \left (1+\frac {x^3}{e}\right )} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = -\frac {3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\sec (c+d x),-\sec (c+d x)\right ) \sqrt [3]{e \sec (c+d x)} \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(749\) vs. \(2(76)=152\).
Time = 7.80 (sec) , antiderivative size = 749, normalized size of antiderivative = 9.86 \[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {720 e \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{6},\frac {2}{3},\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 \left (\frac {1}{2} (c+d x)\right ) (1+\cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{6},\frac {2}{3},\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-\left (4 \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{6},\frac {5}{3},\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 {5}{6},\frac {2}{3},\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 (e \sec (c+d x))^{2/3} \sqrt {a (1+\sec (c+d x))} \left (4320 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{6},\frac {2}{3},\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 \cos ^6\left (\frac {1}{2} (c+d x)\right ) (-1+4 \cos (c+d x))+160 \left (4 \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{6},\frac {5}{3},\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 {5}{6},\frac {2}{3},\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 \cos (c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+12 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{6},\frac {2}{3},\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (20 \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{6},\frac {5}{3},\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+14 \cos (c+d x)+5 \cos (2 (c+d x))-2 \cos (3 (c+d x)))+5 \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{6},\frac {2}{3},\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+14 \cos (c+d x)+5 \cos (2 (c+d x))-2 \cos (3 (c+d x)))-24 \left (40 \operatorname {AppellF1}\left (\frac {5}{2},-\frac {1}{6},\frac {8}{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 )+8 \operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{6},\frac {5}{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 )-5 \operatorname {AppellF1}\left (\frac {5}{2},\frac {11}{6},\frac {2}{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 )\right ) \cos (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]
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\[\int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {1}{3}}}{\sqrt {a +a \sec \left (d x +c \right )}}d x\]
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Timed out. \[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\sqrt [3]{e \sec {\left (c + d x \right )}}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{1/3}}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
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