Integrand size = 27, antiderivative size = 715 \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\frac {3 a \tan (c+d x)}{4 d (e \sec (c+d x))^{4/3} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt [3]{e \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt {a+a \sec (c+d x)} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{16 d e^{5/3} (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}+\frac {5\ 3^{3/4} a^2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{4 \sqrt {2} d e^{5/3} (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}} \]
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Time = 0.71 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3891, 53, 65, 309, 224, 1891} \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\frac {5\ 3^{3/4} a^2 \tan (c+d x) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{4 \sqrt {2} d e^{5/3} (a-a \sec (c+d x)) \sqrt {a \sec (c+d x)+a} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \tan (c+d x) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{16 d e^{5/3} (a-a \sec (c+d x)) \sqrt {a \sec (c+d x)+a} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt {a \sec (c+d x)+a} \sqrt [3]{e \sec (c+d x)}}+\frac {3 a \tan (c+d x)}{4 d \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{4/3}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt {a \sec (c+d x)+a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )} \]
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Rule 53
Rule 65
Rule 224
Rule 309
Rule 1891
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)^{7/3} \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)}{4 d (e \sec (c+d x))^{4/3} \sqrt {a+a \sec (c+d x)}}-\frac {\left (5 a^2 \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(e x)^{4/3} \sqrt {a-a x}} \, dx,x,\sec (c+d x)\right )}{8 d \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{4 d (e \sec (c+d x))^{4/3} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt [3]{e \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {\left (5 a^2 \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{e x} \sqrt {a-a x}} \, dx,x,\sec (c+d x)\right )}{16 d e \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{4 d (e \sec (c+d x))^{4/3} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt [3]{e \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {\left (15 a^2 \tan (c+d x)\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a-\frac {a x^3}{e}}} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{16 d e^2 \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{4 d (e \sec (c+d x))^{4/3} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt [3]{e \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}-\frac {\left (15 a^2 \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-x}{\sqrt {a-\frac {a x^3}{e}}} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{16 d e^2 \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {\left (15 \left (1-\sqrt {3}\right ) 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 )}{16 d e^{5/3} \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = \frac {3 a \tan (c+d x)}{4 d (e \sec (c+d x))^{4/3} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt [3]{e \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {15 a \tan (c+d x)}{8 d e \sqrt {a+a \sec (c+d x)} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 E\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)}{16 d e^{5/3} (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}}}+\frac {5\ 3^{3/4} 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)}{4 \sqrt {2} d e^{5/3} (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 = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{3},\frac {3}{2},1-\sec (c+d x)\right ) \sec ^{\frac {4}{3}}(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{d (e \sec (c+d x))^{4/3}} \]
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\[\int \frac {\sqrt {a +a \sec \left (d x +c \right )}}{\left (e \sec \left (d x +c \right )\right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\int { \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\int \frac {\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\int { \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\int { \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
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