Integrand size = 27, antiderivative size = 78 \[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\frac {3 \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},\sec (c+d x),-\sec (c+d x)\right ) \tan (c+d x)}{2 d \sqrt {1-\sec (c+d x)} (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.21 (sec) , antiderivative size = 78, 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, 524} \[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\frac {3 \tan (c+d x) \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},\sec (c+d x),-\sec (c+d x)\right )}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}} \]
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Rule 129
Rule 524
Rule 3912
Rule 3913
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\sec (c+d x)} \int \frac {1}{(e \sec (c+d x))^{2/3} \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)^{5/3} (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}{x^3 \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 {2}{3},\frac {1}{2},1,\frac {1}{3},\sec (c+d x),-\sec (c+d x)\right ) \tan (c+d x)}{2 d \sqrt {1-\sec (c+d x)} (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(585\) vs. \(2(78)=156\).
Time = 7.85 (sec) , antiderivative size = 585, normalized size of antiderivative = 7.50 \[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sec ^{\frac {7}{6}}(c+d x) \left (-\frac {3}{2} \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{6}}(c+d x) \left (\sin \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {3}{2} (c+d x)\right )\right )+\frac {5 \sqrt {\frac {1}{1+\cos (c+d x)}} (-1+3 \cos (c+d x)) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{5/6} \sin \left (\frac {1}{2} (c+d x)\right ) \left (-3 \cos ^{\frac {5}{6}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\sec ^2\left (\frac {1}{2} (c+d x)\right )}+2 \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 ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{5/6} \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{-120 \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 ) \left (\frac {1}{1+\cos (c+d x)}\right )^{2/3} \left (\frac {\cos (c+d x)}{1+\cos (c+d x)}\right )^{5/6} \sin \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+32 \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 ) \left (\frac {1}{1+\cos (c+d x)}\right )^{2/3} \left (\frac {\cos (c+d x)}{1+\cos (c+d x)}\right )^{5/6} \sin \left (\frac {1}{2} (c+d x)\right ) \tan ^3\left (\frac {1}{2} (c+d x)\right )+5 \sqrt {2} \cos \left (\frac {1}{2} (c+d x)\right ) \left (3-4 \sqrt {2} \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 ) \left (\frac {1}{1+\cos (c+d x)}\right )^{2/3} \left (\frac {\cos (c+d x)}{1+\cos (c+d x)}\right )^{5/6} \tan ^4\left (\frac {1}{2} (c+d x)\right )\right )}\right )}{d (e \sec (c+d x))^{2/3} \sqrt {a (1+\sec (c+d x))}} \]
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\[\int \frac {1}{\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}} \sqrt {a +a \sec \left (d x +c \right )}}d x\]
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Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}\,{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
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