Integrand size = 23, antiderivative size = 383 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=-\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {57 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{80 d (1+\sec (c+d x))}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}-\frac {19\ 3^{3/4} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{80 \sqrt [3]{2} d (1-\sec (c+d x)) (1+\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
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Time = 0.86 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3885, 4086, 3913, 3912, 52, 65, 231} \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=-\frac {19\ 3^{3/4} \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{80 \sqrt [3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 a d}-\frac {9 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 d}+\frac {57 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{80 d (\sec (c+d x)+1)} \]
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Rule 52
Rule 65
Rule 231
Rule 3885
Rule 3912
Rule 3913
Rule 4086
Rubi steps \begin{align*} \text {integral}& = \frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}+\frac {3 \int \sec (c+d x) \left (\frac {5 a}{3}-a \sec (c+d x)\right ) (a+a \sec (c+d x))^{2/3} \, dx}{8 a} \\ & = -\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}+\frac {19}{40} \int \sec (c+d x) (a+a \sec (c+d x))^{2/3} \, dx \\ & = -\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}+\frac {\left (19 (a+a \sec (c+d x))^{2/3}\right ) \int \sec (c+d x) (1+\sec (c+d x))^{2/3} \, dx}{40 (1+\sec (c+d x))^{2/3}} \\ & = -\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}-\frac {\left (19 (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt {1-x}} \, dx,x,\sec (c+d x)\right )}{40 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}} \\ & = -\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {57 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{80 d (1+\sec (c+d x))}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}-\frac {\left (19 (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)^{5/6}} \, dx,x,\sec (c+d x)\right )}{80 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}} \\ & = -\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {57 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{80 d (1+\sec (c+d x))}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}-\frac {\left (57 (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{40 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}} \\ & = -\frac {9 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{40 d}+\frac {57 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{80 d (1+\sec (c+d x))}+\frac {3 (a+a \sec (c+d x))^{5/3} \tan (c+d x)}{8 a d}-\frac {19\ 3^{3/4} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{80 \sqrt [3]{2} d (1-\sec (c+d x)) (1+\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.27 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\frac {(a (1+\sec (c+d x)))^{2/3} \left (38 \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right )+3 \sqrt [6]{1+\sec (c+d x)} \left (2+7 \sec (c+d x)+5 \sec ^2(c+d x)\right )\right ) \tan (c+d x)}{40 d (1+\sec (c+d x))^{7/6}} \]
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\[\int \sec \left (d x +c \right )^{3} \left (a +a \sec \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{3} \,d x } \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}} \sec ^{3}{\left (c + d x \right )}\, dx \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{3} \,d x } \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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