Integrand size = 35, antiderivative size = 317 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {(1015 A-363 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}-\frac {(1887 A-691 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(23 A-11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(109 A-41 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(579 A-199 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}} \] Output:
1/128*(1015*A-363*B)*arctan(1/2*a^(1/2)*sin(d*x+c)*2^(1/2)/cos(d*x+c)^(1/2 )/(a+a*cos(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)*2^(1/2)/a^(7/2 )/d-1/192*(1887*A-691*B)*sec(d*x+c)^(1/2)*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c) )^(1/2)-1/6*(A-B)*sec(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(7/2)-1/4 8*(23*A-11*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(5/2)-1/64* (109*A-41*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(3/2)+1/19 2*(579*A-199*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 11.33 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\frac {i (1015 A-363 B) e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )}{d}-\frac {(21641 A-8469 B+4 (9415 A-3579 B) \cos (c+d x)+8 (3069 A-1145 B) \cos (2 (c+d x))+10164 A \cos (3 (c+d x))-3748 B \cos (3 (c+d x))+1887 A \cos (4 (c+d x))-691 B \cos (4 (c+d x))) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{96 d}\right )}{8 (a (1+\cos (c+d x)))^{7/2}} \] Input:
Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^( 7/2),x]
Output:
(Cos[(c + d*x)/2]^7*((I*(1015*A - 363*B)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I )*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x))) /(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/(d*E^((I/2)*(c + d*x))) - ((216 41*A - 8469*B + 4*(9415*A - 3579*B)*Cos[c + d*x] + 8*(3069*A - 1145*B)*Cos [2*(c + d*x)] + 10164*A*Cos[3*(c + d*x)] - 3748*B*Cos[3*(c + d*x)] + 1887* A*Cos[4*(c + d*x)] - 691*B*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^5*Sec[c + d* x]^(3/2)*Tan[(c + d*x)/2])/(96*d)))/(8*(a*(1 + Cos[c + d*x]))^(7/2))
Time = 2.08 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.09, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3440, 3042, 3457, 27, 3042, 3457, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3261, 218}
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.
| \(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x) (A+B \cos (c+d x))}{(a \cos (c+d x)+a)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3440 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {3 a (5 A-B)-8 a (A-B) \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {3 a (5 A-B)-8 a (A-B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {3 a (5 A-B)-8 a (A-B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int \frac {3 \left (a^2 (63 A-19 B)-2 a^2 (23 A-11 B) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \int \frac {a^2 (63 A-19 B)-2 a^2 (23 A-11 B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \int \frac {a^2 (63 A-19 B)-2 a^2 (23 A-11 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\int \frac {a^3 (579 A-199 B)-4 a^3 (109 A-41 B) \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\int \frac {a^3 (579 A-199 B)-4 a^3 (109 A-41 B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\int \frac {a^3 (579 A-199 B)-4 a^3 (109 A-41 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 \int -\frac {a^4 (1887 A-691 B)-2 a^4 (579 A-199 B) \cos (c+d x)}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{3 a}+\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^4 (1887 A-691 B)-2 a^4 (579 A-199 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^4 (1887 A-691 B)-2 a^4 (579 A-199 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 \int -\frac {3 a^5 (1015 A-363 B)}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}+\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-3 a^4 (1015 A-363 B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-3 a^4 (1015 A-363 B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {6 a^5 (1015 A-363 B) \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}+\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {3 \sqrt {2} a^{7/2} (1015 A-363 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\) |
Input:
Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(7/2),x ]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/6*((A - B)*Sin[c + d*x])/(d*Cos[ c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) + (-1/4*(a*(23*A - 11*B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) + (3*(-1/2*(a^2* (109*A - 41*B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3 /2)) + ((2*a^3*(579*A - 199*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[ a + a*Cos[c + d*x]]) - ((-3*Sqrt[2]*a^(7/2)*(1015*A - 363*B)*ArcTan[(Sqrt[ a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^4*(1887*A - 691*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a* Cos[c + d*x]]))/(3*a))/(4*a^2)))/(8*a^2))/(12*a^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* (x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p Int[(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g , m, n, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && !(IntegerQ[m] && I ntegerQ[n])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Time = 14.84 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \sqrt {a \left (\cos \left (d x +c \right )+1\right )}\, \sec \left (d x +c \right )^{\frac {5}{2}} \left (A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (3045 \cos \left (d x +c \right )^{6}+12180 \cos \left (d x +c \right )^{5}+18270 \cos \left (d x +c \right )^{4}+12180 \cos \left (d x +c \right )^{3}+3045 \cos \left (d x +c \right )^{2}\right )+B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (-1089 \cos \left (d x +c \right )^{6}-4356 \cos \left (d x +c \right )^{5}-6534 \cos \left (d x +c \right )^{4}-4356 \cos \left (d x +c \right )^{3}-1089 \cos \left (d x +c \right )^{2}\right )+\cos \left (d x +c \right ) \sin \left (d x +c \right ) \left (1887 \cos \left (d x +c \right )^{4}+5082 \cos \left (d x +c \right )^{3}+4251 \cos \left (d x +c \right )^{2}+896 \cos \left (d x +c \right )-128\right ) \sqrt {2}\, A +\cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) \left (-691 \cos \left (d x +c \right )^{3}-1874 \cos \left (d x +c \right )^{2}-1599 \cos \left (d x +c \right )-384\right ) \sqrt {2}\, B \right )}{384 d \left (\cos \left (d x +c \right )+1\right ) \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) a^{4}}\) | \(355\) |
| parts | \(-\frac {A \sqrt {2}\, \sqrt {a \left (\cos \left (d x +c \right )+1\right )}\, \sec \left (d x +c \right )^{\frac {5}{2}} \left (\cos \left (d x +c \right ) \sin \left (d x +c \right ) \left (1887 \cos \left (d x +c \right )^{4}+5082 \cos \left (d x +c \right )^{3}+4251 \cos \left (d x +c \right )^{2}+896 \cos \left (d x +c \right )-128\right ) \sqrt {2}+\sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (3045 \cos \left (d x +c \right )^{6}+12180 \cos \left (d x +c \right )^{5}+18270 \cos \left (d x +c \right )^{4}+12180 \cos \left (d x +c \right )^{3}+3045 \cos \left (d x +c \right )^{2}\right )\right )}{384 d \left (\cos \left (d x +c \right )+1\right ) \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) a^{4}}+\frac {B \cos \left (d x +c \right )^{2} \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\sin \left (d x +c \right ) \left (691 \cos \left (d x +c \right )^{3}+1874 \cos \left (d x +c \right )^{2}+1599 \cos \left (d x +c \right )+384\right ) \sqrt {2}+\left (1089 \cos \left (d x +c \right )^{4}+4356 \cos \left (d x +c \right )^{3}+6534 \cos \left (d x +c \right )^{2}+4356 \cos \left (d x +c \right )+1089\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sec \left (d x +c \right )^{\frac {5}{2}}}{192 d \left (\cos \left (d x +c \right )+1\right ) \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) a^{4}}\) | \(415\) |
Input:
int((A+B*cos(d*x+c))*sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x,method=_RET URNVERBOSE)
Output:
-1/384/d*2^(1/2)*(a*(cos(d*x+c)+1))^(1/2)*sec(d*x+c)^(5/2)/(cos(d*x+c)+1)/ (cos(d*x+c)^3+3*cos(d*x+c)^2+3*cos(d*x+c)+1)*(A*arcsin(cot(d*x+c)-csc(d*x+ c))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(3045*cos(d*x+c)^6+12180*cos(d*x+c)^ 5+18270*cos(d*x+c)^4+12180*cos(d*x+c)^3+3045*cos(d*x+c)^2)+B*arcsin(cot(d* x+c)-csc(d*x+c))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-1089*cos(d*x+c)^6-435 6*cos(d*x+c)^5-6534*cos(d*x+c)^4-4356*cos(d*x+c)^3-1089*cos(d*x+c)^2)+cos( d*x+c)*sin(d*x+c)*(1887*cos(d*x+c)^4+5082*cos(d*x+c)^3+4251*cos(d*x+c)^2+8 96*cos(d*x+c)-128)*2^(1/2)*A+cos(d*x+c)^2*sin(d*x+c)*(-691*cos(d*x+c)^3-18 74*cos(d*x+c)^2-1599*cos(d*x+c)-384)*2^(1/2)*B)/a^4
Time = 0.14 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {3 \, \sqrt {2} {\left ({\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left ({\left (1887 \, A - 691 \, B\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2541 \, A - 937 \, B\right )} \cos \left (d x + c\right )^{3} + 39 \, {\left (109 \, A - 41 \, B\right )} \cos \left (d x + c\right )^{2} + 128 \, {\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right ) - 128 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \] Input:
integrate((A+B*cos(d*x+c))*sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x, algo rithm="fricas")
Output:
-1/384*(3*sqrt(2)*((1015*A - 363*B)*cos(d*x + c)^5 + 4*(1015*A - 363*B)*co s(d*x + c)^4 + 6*(1015*A - 363*B)*cos(d*x + c)^3 + 4*(1015*A - 363*B)*cos( d*x + c)^2 + (1015*A - 363*B)*cos(d*x + c))*sqrt(a)*arctan(sqrt(2)*sqrt(a* cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) + 2*((1887*A - 691*B)*cos(d*x + c)^4 + 2*(2541*A - 937*B)*cos(d*x + c)^3 + 39*(109*A - 41*B)*cos(d*x + c)^2 + 128*(7*A - 3*B)*cos(d*x + c) - 128*A)*sqrt(a*cos(d* x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^4*d*cos(d*x + c)^5 + 4*a^4 *d*cos(d*x + c)^4 + 6*a^4*d*cos(d*x + c)^3 + 4*a^4*d*cos(d*x + c)^2 + a^4* d*cos(d*x + c))
Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c))*sec(d*x+c)**(5/2)/(a+a*cos(d*x+c))**(7/2),x)
Output:
Timed out
Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c))*sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x, algo rithm="maxima")
Output:
Timed out
\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c))*sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x, algo rithm="giac")
Output:
integrate((B*cos(d*x + c) + A)*sec(d*x + c)^(5/2)/(a*cos(d*x + c) + a)^(7/ 2), x)
Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:
int(((A + B*cos(c + d*x))*(1/cos(c + d*x))^(5/2))/(a + a*cos(c + d*x))^(7/ 2),x)
Output:
int(((A + B*cos(c + d*x))*(1/cos(c + d*x))^(5/2))/(a + a*cos(c + d*x))^(7/ 2), x)
\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1}d x \right ) a \right )}{a^{4}} \] Input:
int((A+B*cos(d*x+c))*sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x)
Output:
(sqrt(a)*(int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1)*cos(c + d*x)*sec( c + d*x)**2)/(cos(c + d*x)**4 + 4*cos(c + d*x)**3 + 6*cos(c + d*x)**2 + 4* cos(c + d*x) + 1),x)*b + int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1)*se c(c + d*x)**2)/(cos(c + d*x)**4 + 4*cos(c + d*x)**3 + 6*cos(c + d*x)**2 + 4*cos(c + d*x) + 1),x)*a))/a**4