Integrand size = 31, antiderivative size = 165 \[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=-\frac {3 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1-3 m),\frac {1}{6} (7-3 m),\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) \sqrt {\sin ^2(c+d x)}}+\frac {3 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-2-3 m),\frac {1}{6} (4-3 m),\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt {\sin ^2(c+d x)}} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {20, 3872, 3857, 2722} \[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\frac {3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-3 m-2),\frac {1}{6} (4-3 m),\cos ^2(c+d x)\right )}{d (3 m+2) \sqrt {\sin ^2(c+d x)}}-\frac {3 A \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1-3 m),\frac {1}{6} (7-3 m),\cos ^2(c+d x)\right )}{d (1-3 m) \sqrt {\sin ^2(c+d x)}} \]
[In]
[Out]
Rule 20
Rule 2722
Rule 3857
Rule 3872
Rubi steps \begin{align*} \text {integral}& = \frac {(b \sec (c+d x))^{2/3} \int \sec ^{\frac {2}{3}+m}(c+d x) (A+B \sec (c+d x)) \, dx}{\sec ^{\frac {2}{3}}(c+d x)} \\ & = \frac {\left (A (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac {2}{3}+m}(c+d x) \, dx}{\sec ^{\frac {2}{3}}(c+d x)}+\frac {\left (B (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac {5}{3}+m}(c+d x) \, dx}{\sec ^{\frac {2}{3}}(c+d x)} \\ & = \left (A \cos ^{\frac {2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac {2}{3}-m}(c+d x) \, dx+\left (B \cos ^{\frac {2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac {5}{3}-m}(c+d x) \, dx \\ & = -\frac {3 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1-3 m),\frac {1}{6} (7-3 m),\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) \sqrt {\sin ^2(c+d x)}}+\frac {3 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-2-3 m),\frac {1}{6} (4-3 m),\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.85 \[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\frac {3 \csc (c+d x) \left (A (5+3 m) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (2+3 m),\frac {1}{6} (8+3 m),\sec ^2(c+d x)\right )+B (2+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\sec ^2(c+d x)\right )\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sqrt {-\tan ^2(c+d x)}}{d (2+3 m) (5+3 m)} \]
[In]
[Out]
\[\int \sec \left (d x +c \right )^{m} \left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \sec \left (d x +c \right )\right )d x\]
[In]
[Out]
\[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m} \,d x } \]
[In]
[Out]
\[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\int \left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}} \left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m} \,d x } \]
[In]
[Out]
\[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \]
[In]
[Out]