Integrand size = 45, antiderivative size = 353 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {a^{5/2} (1304 A+1132 B+1015 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{512 d}+\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)} \]
1/60*a*(12*B+5*C)*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(7/2)+1/6 *C*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d/cos(d*x+c)^(7/2)+1/512*a^(5/2)*(130 4*A+1132*B+1015*C)*arcsinh(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))*cos( d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+1/960*a^3*(680*A+628*B+545*C)*sin(d*x+c)/d /cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^(1/2)+1/768*a^3*(1304*A+1132*B+1015*C)* sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(1/2)+1/512*a^3*(1304*A+113 2*B+1015*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(1/2)+1/480*a^2 *(120*A+156*B+115*C)*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(7/2)
Leaf count is larger than twice the leaf count of optimal. \(707\) vs. \(2(353)=706\).
Time = 15.21 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.00 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {4 \sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )} \left (\frac {163 A \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{512 \sqrt {2}}+\frac {283 B \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{1024 \sqrt {2}}+\frac {1015 C \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{4096 \sqrt {2}}+\frac {C \sin \left (\frac {1}{2} (c+d x)\right )}{48 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {B \sin \left (\frac {1}{2} (c+d x)\right )}{40 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {7 C \sin \left (\frac {1}{2} (c+d x)\right )}{96 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {A \sin \left (\frac {1}{2} (c+d x)\right )}{32 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {29 B \sin \left (\frac {1}{2} (c+d x)\right )}{320 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {29 C \sin \left (\frac {1}{2} (c+d x)\right )}{256 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {23 A \sin \left (\frac {1}{2} (c+d x)\right )}{192 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {283 B \sin \left (\frac {1}{2} (c+d x)\right )}{1920 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {203 C \sin \left (\frac {1}{2} (c+d x)\right )}{1536 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {163 A \sin \left (\frac {1}{2} (c+d x)\right )}{768 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {283 B \sin \left (\frac {1}{2} (c+d x)\right )}{1536 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {1015 C \sin \left (\frac {1}{2} (c+d x)\right )}{6144 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {163 A \sin \left (\frac {1}{2} (c+d x)\right )}{512 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {283 B \sin \left (\frac {1}{2} (c+d x)\right )}{1024 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {1015 C \sin \left (\frac {1}{2} (c+d x)\right )}{4096 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]
Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2))/Cos[c + d*x]^(5/2),x]
(4*Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*(A + B*Sec[c + d*x] + C *Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((163*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])/(512*Sqrt[2]) + (28 3*B*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])/(1024*Sqrt[2]) + (1015*C*ArcTanh[Sq rt[2]*Sin[(c + d*x)/2]])/(4096*Sqrt[2]) + (C*Sin[(c + d*x)/2])/(48*(1 - 2* Sin[(c + d*x)/2]^2)^6) + (B*Sin[(c + d*x)/2])/(40*(1 - 2*Sin[(c + d*x)/2]^ 2)^5) + (7*C*Sin[(c + d*x)/2])/(96*(1 - 2*Sin[(c + d*x)/2]^2)^5) + (A*Sin[ (c + d*x)/2])/(32*(1 - 2*Sin[(c + d*x)/2]^2)^4) + (29*B*Sin[(c + d*x)/2])/ (320*(1 - 2*Sin[(c + d*x)/2]^2)^4) + (29*C*Sin[(c + d*x)/2])/(256*(1 - 2*S in[(c + d*x)/2]^2)^4) + (23*A*Sin[(c + d*x)/2])/(192*(1 - 2*Sin[(c + d*x)/ 2]^2)^3) + (283*B*Sin[(c + d*x)/2])/(1920*(1 - 2*Sin[(c + d*x)/2]^2)^3) + (203*C*Sin[(c + d*x)/2])/(1536*(1 - 2*Sin[(c + d*x)/2]^2)^3) + (163*A*Sin[ (c + d*x)/2])/(768*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (283*B*Sin[(c + d*x)/2] )/(1536*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (1015*C*Sin[(c + d*x)/2])/(6144*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (163*A*Sin[(c + d*x)/2])/(512*(1 - 2*Sin[(c + d*x)/2]^2)) + (283*B*Sin[(c + d*x)/2])/(1024*(1 - 2*Sin[(c + d*x)/2]^2)) + (1015*C*Sin[(c + d*x)/2])/(4096*(1 - 2*Sin[(c + d*x)/2]^2))))/(d*(A + 2 *C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))
Time = 2.25 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 4753, 3042, 4576, 27, 3042, 4506, 27, 3042, 4506, 27, 3042, 4504, 3042, 4290, 3042, 4290, 3042, 4288, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^{5/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )}{\cos (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )dx\) |
| \(\Big \downarrow \) 4576 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {1}{2} \sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{5/2} (a (12 A+5 C)+a (12 B+5 C) \sec (c+d x))dx}{6 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{5/2} (a (12 A+5 C)+a (12 B+5 C) \sec (c+d x))dx}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (12 A+5 C)+a (12 B+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{5} \int \frac {1}{2} \sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{3/2} \left (15 (8 A+4 B+5 C) a^2+(120 A+156 B+115 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \int \sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{3/2} \left (15 (8 A+4 B+5 C) a^2+(120 A+156 B+115 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (15 (8 A+4 B+5 C) a^2+(120 A+156 B+115 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sec (c+d x) a+a} \left (5 (312 A+252 B+235 C) a^3+3 (680 A+628 B+545 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sec (c+d x) a+a} \left (5 (312 A+252 B+235 C) a^3+3 (680 A+628 B+545 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (5 (312 A+252 B+235 C) a^3+3 (680 A+628 B+545 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 4504 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 4290 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \left (\frac {3}{4} \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 4290 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 4288 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \left (\frac {3}{4} \left (\frac {a \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {1}{\sqrt {\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\right )+\frac {a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {a^2 (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac {1}{10} \left (\frac {a^3 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}+\frac {1}{8} \left (\frac {a^4 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}+\frac {5}{2} a^3 (1304 A+1132 B+1015 C) \left (\frac {3}{4} \left (\frac {\sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )\right )\right )}{12 a}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\right )\) |
Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/C os[c + d*x]^(5/2),x]
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d) + ((a^2*(12*B + 5*C)*Sec[c + d*x]^(7/2)* (a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + ((a^3*(120*A + 156*B + 11 5*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + ((a ^4*(680*A + 628*B + 545*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*Sqrt[a + a* Sec[c + d*x]]) + (5*a^3*(1304*A + 1132*B + 1015*C)*((a*Sec[c + d*x]^(5/2)* Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]]) + (3*((Sqrt[a]*ArcSinh[(Sqrt[ a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sec[c + d*x]^(3/2)*Sin[ c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])))/4))/2)/8)/10)/(12*a))
3.13.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(a/(b*f))*Sqrt[a*(d/b)] Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; FreeQ[{a , b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*(d/b), 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*d*Cot[e + f*x]*((d*Csc[e + f*x])^(n - 1)/( f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[2*a*d*((n - 1)/(b*(2*n - 1))) Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1), x], x] /; Fre eQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[-2*b*B*C ot[e + f*x]*((d*Csc[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[(A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1)) Int[Sqrt[a + b*Csc[e + f* x]]*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ [A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && !LtQ[n, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Cs c[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b* B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m , n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs. \(2(305)=610\).
Time = 1.17 (sec) , antiderivative size = 825, normalized size of antiderivative = 2.34
int((a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2 ),x,method=_RETURNVERBOSE)
1/15360*a^2/d*(19560*A*cos(d*x+c)^6*arctan(1/2*(cos(d*x+c)-sin(d*x+c)+1)/( 1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))-19560*A*arctan(1/2*(cos(d*x+c)+si n(d*x+c)+1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^6+39120*A *(-1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^5+16980*B*cos(d*x+c)^6*ar ctan(1/2*(cos(d*x+c)-sin(d*x+c)+1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2 ))-16980*B*cos(d*x+c)^6*arctan(1/2*(cos(d*x+c)+sin(d*x+c)+1)/(1+cos(d*x+c) )/(-1/(1+cos(d*x+c)))^(1/2))+33960*B*cos(d*x+c)^5*sin(d*x+c)*(-1/(1+cos(d* x+c)))^(1/2)+15225*C*cos(d*x+c)^6*arctan(1/2*(cos(d*x+c)-sin(d*x+c)+1)/(1+ cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))-15225*C*arctan(1/2*(cos(d*x+c)+sin( d*x+c)+1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^6+30450*C*( -1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^5+26080*A*cos(d*x+c)^4*sin( d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+22640*B*sin(d*x+c)*cos(d*x+c)^4*(-1/(1+co s(d*x+c)))^(1/2)+20300*C*cos(d*x+c)^4*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2) +14720*A*cos(d*x+c)^3*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+18112*B*sin(d*x +c)*cos(d*x+c)^3*(-1/(1+cos(d*x+c)))^(1/2)+16240*C*cos(d*x+c)^3*sin(d*x+c) *(-1/(1+cos(d*x+c)))^(1/2)+3840*A*sin(d*x+c)*cos(d*x+c)^2*(-1/(1+cos(d*x+c )))^(1/2)+11136*B*cos(d*x+c)^2*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+13920* C*sin(d*x+c)*cos(d*x+c)^2*(-1/(1+cos(d*x+c)))^(1/2)+3072*B*cos(d*x+c)*sin( d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+8960*C*cos(d*x+c)*sin(d*x+c)*(-1/(1+cos(d *x+c)))^(1/2)+2560*C*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2))*(a*(1+sec(d*...
Time = 0.53 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.77 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\left [\frac {4 \, {\left (15 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \, {\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 15 \, {\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 4 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{30720 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}, \frac {2 \, {\left (15 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \, {\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 15 \, {\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{15360 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}\right ] \]
integrate((a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c )^(5/2),x, algorithm="fricas")
[1/30720*(4*(15*(1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^5 + 10*(1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^4 + 8*(920*A + 1132*B + 1015*C)*a^2*c os(d*x + c)^3 + 48*(40*A + 116*B + 145*C)*a^2*cos(d*x + c)^2 + 128*(12*B + 35*C)*a^2*cos(d*x + c) + 1280*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 15*((1304*A + 1132*B + 1015*C)*a^2*c os(d*x + c)^7 + (1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^6)*sqrt(a)*log ((a*cos(d*x + c)^3 - 4*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(co s(d*x + c) - 2)*sqrt(cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c)^2 + 8*a )/(cos(d*x + c)^3 + cos(d*x + c)^2)))/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6 ), 1/15360*(2*(15*(1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^5 + 10*(1304 *A + 1132*B + 1015*C)*a^2*cos(d*x + c)^4 + 8*(920*A + 1132*B + 1015*C)*a^2 *cos(d*x + c)^3 + 48*(40*A + 116*B + 145*C)*a^2*cos(d*x + c)^2 + 128*(12*B + 35*C)*a^2*cos(d*x + c) + 1280*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 15*((1304*A + 1132*B + 1015*C)*a^2 *cos(d*x + c)^7 + (1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^6)*sqrt(-a)* arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c ))*sin(d*x + c)/(a*cos(d*x + c)^2 - a*cos(d*x + c) - 2*a)))/(d*cos(d*x + c )^7 + d*cos(d*x + c)^6)]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 16461 vs. \(2 (305) = 610\).
Time = 2.62 (sec) , antiderivative size = 16461, normalized size of antiderivative = 46.63 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Too large to display} \]
integrate((a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c )^(5/2),x, algorithm="maxima")
-1/30720*(40*(1956*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c)) *cos(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 652*(sqrt(2)*a^2* sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d* x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(13/4*arctan2(sin(2*d*x + 2* c), cos(2*d*x + 2*c))) + 6204*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^ 2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2* d*x + 2*c))*cos(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2060*( sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)* a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin (2*d*x + 2*c), cos(2*d*x + 2*c))) + 2060*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4 *sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2) *a^2*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)) ) - 6204*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(5/4*a rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 652*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d* x + 2*c))) - 1956*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c...
\[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c )^(5/2),x, algorithm="giac")
Timed out. \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x \]
int(((a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/c os(c + d*x)^(5/2),x)