Integrand size = 27, antiderivative size = 791 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=-\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac {3 \sqrt {2} A \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt {1-\sec (c+d x)}}-\frac {3 \left (1+\sqrt {3}\right ) (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{a d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {3 \sqrt [3]{2} \sqrt [4]{3} (A+2 C) E\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 ) \sqrt [3]{a+a \sec (c+d x)} \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)}{a d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \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}}}+\frac {3^{3/4} \left (1-\sqrt {3}\right ) (A+2 C) \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 ) \sqrt [3]{a+a \sec (c+d x)} \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)}{2^{2/3} a d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \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 = 1.07 (sec) , antiderivative size = 791, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4138, 4009, 3864, 3863, 141, 3913, 3912, 65, 314, 231, 1895} \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\frac {3 \sqrt {2} A \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{5 a d \sqrt {1-\sec (c+d x)}}+\frac {3^{3/4} \left (1-\sqrt {3}\right ) (A+2 C) \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}} \sqrt [3]{a \sec (c+d x)+a} \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 )}{2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \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 \sqrt [3]{2} \sqrt [4]{3} (A+2 C) \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}} \sqrt [3]{a \sec (c+d x)+a} E\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 )}{a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \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 (A+C) \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-\frac {3 \left (1+\sqrt {3}\right ) (A+2 C) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{a d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )} \]
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Rule 65
Rule 141
Rule 231
Rule 314
Rule 1895
Rule 3863
Rule 3864
Rule 3912
Rule 3913
Rule 4009
Rule 4138
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}-\frac {3 \int \sqrt [3]{a+a \sec (c+d x)} \left (-\frac {a A}{3}-\frac {1}{3} a (A+2 C) \sec (c+d x)\right ) \, dx}{a^2} \\ & = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac {A \int \sqrt [3]{a+a \sec (c+d x)} \, dx}{a}+\frac {(A+2 C) \int \sec (c+d x) \sqrt [3]{a+a \sec (c+d x)} \, dx}{a} \\ & = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac {\left (A \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sqrt [3]{1+\sec (c+d x)} \, dx}{a \sqrt [3]{1+\sec (c+d x)}}+\frac {\left ((A+2 C) \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \, dx}{a \sqrt [3]{1+\sec (c+d x)}} \\ & = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}-\frac {\left (A \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{a d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}-\frac {\left ((A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{a d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}} \\ & = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac {3 \sqrt {2} A \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt {1-\sec (c+d x)}}-\frac {\left (6 (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{a d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}} \\ & = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac {3 \sqrt {2} A \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt {1-\sec (c+d x)}}+\frac {\left (3 (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{a d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}+\frac {\left (3\ 2^{2/3} \left (1-\sqrt {3}\right ) (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \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 )}{a d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}} \\ & = -\frac {3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac {3 \sqrt {2} A \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt {1-\sec (c+d x)}}-\frac {3 \left (1+\sqrt {3}\right ) (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{a d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {3 \sqrt [3]{2} \sqrt [4]{3} (A+2 C) E\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 ) \sqrt [3]{a+a \sec (c+d x)} \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)}{a d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \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}}}+\frac {3^{3/4} \left (1-\sqrt {3}\right ) (A+2 C) \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 ) \sqrt [3]{a+a \sec (c+d x)} \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)}{2^{2/3} a d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \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*}
\[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx \]
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\[\int \frac {A +C \sec \left (d x +c \right )^{2}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
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