Integrand size = 33, antiderivative size = 92 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{7 d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {16, 3091, 2722} \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \sin (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(c+d x)\right )}{7 d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
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Rule 16
Rule 2722
Rule 3091
Rubi steps \begin{align*} \text {integral}& = b^3 \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{10/3}} \, dx \\ & = \frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {1}{7} (b (4 A+7 C)) \int \frac {1}{(b \cos (c+d x))^{4/3}} \, dx \\ & = \frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{7 d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.99 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 \left (7 C \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(c+d x)\right ) \sin (c+d x)+A \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {1}{2},-\frac {1}{6},\cos ^2(c+d x)\right ) \sec (c+d x) \tan (c+d x)\right )}{7 d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
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\[\int \frac {\left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \]
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