Optimal. Leaf size=223 \[ \frac{2 (A (2 n+7)+C (2 n+5)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (-2 n-3),\frac{1}{4} (1-2 n),\cos ^2(c+d x)\right )}{d (2 n+3) (2 n+7) \sqrt{\sin ^2(c+d x)}}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (-2 n-5),\frac{1}{4} (-2 n-1),\cos ^2(c+d x)\right )}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+7)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.195954, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {20, 4047, 3772, 2643, 4046} \[ \frac{2 (A (2 n+7)+C (2 n+5)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-2 n-3);\frac{1}{4} (1-2 n);\cos ^2(c+d x)\right )}{d (2 n+3) (2 n+7) \sqrt{\sin ^2(c+d x)}}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-2 n-5);\frac{1}{4} (-2 n-1);\cos ^2(c+d x)\right )}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+7)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac{5}{2}+n}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac{5}{2}+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx+\left (B \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac{7}{2}+n}(c+d x) \, dx\\ &=\frac{2 C \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (7+2 n)}+\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{7}{2}-n}(c+d x) \, dx+\frac{\left (\left (C \left (\frac{5}{2}+n\right )+A \left (\frac{7}{2}+n\right )\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac{5}{2}+n}(c+d x) \, dx}{\frac{7}{2}+n}\\ &=\frac{2 C \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (7+2 n)}+\frac{2 B \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-5-2 n);\frac{1}{4} (-1-2 n);\cos ^2(c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n) \sqrt{\sin ^2(c+d x)}}+\frac{\left (\left (C \left (\frac{5}{2}+n\right )+A \left (\frac{7}{2}+n\right )\right ) \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{7}{2}+n}\\ &=\frac{2 C \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (7+2 n)}+\frac{2 (C (5+2 n)+A (7+2 n)) \, _2F_1\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) (7+2 n) \sqrt{\sin ^2(c+d x)}}+\frac{2 B \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-5-2 n);\frac{1}{4} (-1-2 n);\cos ^2(c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.87678, size = 493, normalized size = 2.21 \[ -\frac{i 2^{n+\frac{9}{2}} e^{2 i c-\frac{1}{2} i d (2 n+1) x} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac{1}{2}} \left (1+e^{2 i (c+d x)}\right )^{n+\frac{1}{2}} \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (e^{2 i c} \left (\frac{2 (A+2 C) e^{\frac{1}{2} i d (2 n+9) x} \text{Hypergeometric2F1}\left (n+\frac{9}{2},\frac{1}{4} (2 n+9),\frac{1}{4} (2 n+13),-e^{2 i (c+d x)}\right )}{2 n+9}+\frac{A e^{\frac{1}{2} i (4 c+d (2 n+13) x)} \text{Hypergeometric2F1}\left (n+\frac{9}{2},\frac{1}{4} (2 n+13),\frac{1}{4} (2 n+17),-e^{2 i (c+d x)}\right )}{2 n+13}+\frac{2 B e^{\frac{1}{2} i (2 c+d (2 n+11) x)} \text{Hypergeometric2F1}\left (n+\frac{9}{2},\frac{1}{4} (2 n+11),\frac{1}{4} (2 n+15),-e^{2 i (c+d x)}\right )}{2 n+11}\right )+\frac{A e^{\frac{1}{2} i d (2 n+5) x} \text{Hypergeometric2F1}\left (n+\frac{9}{2},\frac{1}{4} (2 n+5),\frac{1}{4} (2 n+9),-e^{2 i (c+d x)}\right )}{2 n+5}+\frac{2 B e^{\frac{1}{2} i (2 c+d (2 n+7) x)} \text{Hypergeometric2F1}\left (n+\frac{9}{2},\frac{1}{4} (2 n+7),\frac{1}{4} (2 n+11),-e^{2 i (c+d x)}\right )}{2 n+7}\right )}{d (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.236, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+B\sec \left ( dx+c \right ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \sec \left (d x + c\right )^{4} + B \sec \left (d x + c\right )^{3} + A \sec \left (d x + c\right )^{2}\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sqrt{\sec \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + 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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]