Integrand size = 25, antiderivative size = 327 \[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=-\frac {3 a \sec ^{\frac {5}{3}}(c+d x) \sin (c+d x)}{2 d \sqrt [3]{a (1+\sec (c+d x))}}+\frac {9 \sec ^{\frac {2}{3}}(c+d x) (a (1+\sec (c+d x)))^{2/3} \sin (c+d x)}{4 d}-\frac {9 (a (1+\sec (c+d x)))^{2/3} \tan (c+d x)}{4 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{7/3}}+\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {5}{4},\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} (a (1+\sec (c+d x)))^{2/3} \tan (c+d x)}{8 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{4/3}}-\frac {5 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {7}{4},\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} (a (1+\sec (c+d x)))^{2/3} \tan ^3(c+d x)}{8 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{10/3}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3913, 3910, 138} \[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {2}{3},-\frac {1}{6},\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (\sec (c+d x)+1)^{7/6}} \]
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Rule 138
Rule 3910
Rule 3913
Rubi steps \begin{align*} \text {integral}& = \frac {(a+a \sec (c+d x))^{2/3} \int \sec ^{\frac {5}{3}}(c+d x) (1+\sec (c+d x))^{2/3} \, dx}{(1+\sec (c+d x))^{2/3}} \\ & = \frac {\left ((a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(1-x)^{2/3} \sqrt [6]{2-x}}{\sqrt {x}} \, dx,x,1-\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}} \\ & = \frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {2}{3},-\frac {1}{6},\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{d (1+\sec (c+d x))^{7/6}} \\ \end{align*}
Time = 9.27 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.84 \[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\frac {(a (1+\sec (c+d x)))^{2/3} \left (-3 \sec ^3\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \left (\sin \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {3}{2} (c+d x)\right )\right )+\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {5}{4},\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} \sqrt [3]{\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right )-5 \sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {7}{4},\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} \sqrt [3]{\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \tan ^3\left (\frac {1}{2} (c+d x)\right )\right )}{8 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{2/3}} \]
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\[\int \sec \left (d x +c \right )^{\frac {5}{3}} \left (a +a \sec \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
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\[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {5}{3}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {5}{3}} \,d x } \]
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\[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {5}{3}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/3} \,d x \]
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