Integrand size = 25, antiderivative size = 90 \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=-\frac {3 b (2 A-C) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{8 d (b \sec (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}}+\frac {3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}} \]
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Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4131, 3857, 2722} \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\frac {3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}-\frac {3 b (2 A-C) \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right )}{8 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
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Rule 2722
Rule 3857
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}+\frac {1}{2} (2 A-C) \int \frac {1}{\sqrt [3]{b \sec (c+d x)}} \, dx \\ & = \frac {3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}+\frac {1}{2} \left ((2 A-C) \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \sqrt [3]{\frac {\cos (c+d x)}{b}} \, dx \\ & = -\frac {3 (2 A-C) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{8 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\frac {3 C \tan (c+d x)-3 (2 A-C) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{2 d \sqrt [3]{b \sec (c+d x)}} \]
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\[\int \frac {A +C \sec \left (d x +c \right )^{2}}{\left (b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\sqrt [3]{b \sec {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]
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