Integrand size = 31, antiderivative size = 163 \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\frac {2 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1-2 n),\frac {1}{4} (3-2 n),\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt {\sin ^2(c+d x)}}+\frac {2 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-3-2 n),\frac {1}{4} (1-2 n),\cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (3+2 n) \sqrt {\sin ^2(c+d x)}} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 163, 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} \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-2 n-1),\frac {1}{4} (3-2 n),\cos ^2(c+d x)\right )}{d (2 n+1) \sqrt {\sin ^2(c+d x)}}+\frac {2 B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-2 n-3),\frac {1}{4} (1-2 n),\cos ^2(c+d x)\right )}{d (2 n+3) \sqrt {\sin ^2(c+d x)}} \]
[In]
[Out]
Rule 20
Rule 2722
Rule 3857
Rule 3872
Rubi steps \begin{align*} \text {integral}& = \left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac {3}{2}+n}(c+d x) (A+B \sec (c+d x)) \, dx \\ & = \left (A \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac {3}{2}+n}(c+d x) \, dx+\left (B \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac {5}{2}+n}(c+d x) \, dx \\ & = \left (A \cos ^{\frac {1}{2}+n}(c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{-\frac {3}{2}-n}(c+d x) \, dx+\left (B \cos ^{\frac {1}{2}+n}(c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{-\frac {5}{2}-n}(c+d x) \, dx \\ & = \frac {2 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1-2 n),\frac {1}{4} (3-2 n),\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt {\sin ^2(c+d x)}}+\frac {2 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-3-2 n),\frac {1}{4} (1-2 n),\cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (3+2 n) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86 \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\frac {2 \csc (c+d x) \left (A (5+2 n) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (3+2 n),\frac {1}{4} (7+2 n),\sec ^2(c+d x)\right )+B (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\sec ^2(c+d x)\right )\right ) \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \sqrt {-\tan ^2(c+d x)}}{d (3+2 n) (5+2 n)} \]
[In]
[Out]
\[\int \sec \left (d x +c \right )^{\frac {3}{2}} \left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )\right )d x\]
[In]
[Out]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
[In]
[Out]