Integrand size = 27, antiderivative size = 326 \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} a^2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{2 d e (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}} \]
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Time = 0.37 (sec) , antiderivative size = 326, 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 = {3891, 53, 65, 224} \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} a^2 \tan (c+d x) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{2 d e (a-a \sec (c+d x)) \sqrt {a \sec (c+d x)+a} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}+\frac {3 a \tan (c+d x)}{2 d \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}} \]
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Rule 53
Rule 65
Rule 224
Rule 3891
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 e \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(e x)^{5/3} \sqrt {a-a x}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}-\frac {\left (a^2 \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(e x)^{2/3} \sqrt {a-a x}} \, dx,x,\sec (c+d x)\right )}{4 d \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}-\frac {\left (3 a^2 \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {a x^3}{e}}} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{4 d e \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} a^2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{2 d e (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {3}{2},1-\sec (c+d x)\right ) \sec ^{\frac {2}{3}}(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{d (e \sec (c+d x))^{2/3}} \]
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\[\int \frac {\sqrt {a +a \sec \left (d x +c \right )}}{\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\int { \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\int \frac {\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\int { \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\int { \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
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