Integrand size = 45, antiderivative size = 352 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {(8 A-20 B+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{4 a^{5/2} d}-\frac {(43 A-115 B+219 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B+C) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {(3 A-11 B+19 C) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(7 A-15 B+31 C) \sin (c+d x)}{16 a^2 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}-\frac {(11 A-35 B+63 C) \sin (c+d x)}{16 a^2 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \] Output:
1/4*(8*A-20*B+39*C)*arcsin(a^(1/2)*sin(d*x+c)/(a+a*cos(d*x+c))^(1/2))*cos( d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^(5/2)/d-1/32*(43*A-115*B+219*C)*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^(5/2)/d-1/4*(A-B+C)*sin(d*x+c)/d/(a +a*cos(d*x+c))^(5/2)/sec(d*x+c)^(7/2)-1/16*(3*A-11*B+19*C)*sin(d*x+c)/a/d/ (a+a*cos(d*x+c))^(3/2)/sec(d*x+c)^(5/2)+1/16*(7*A-15*B+31*C)*sin(d*x+c)/a^ 2/d/(a+a*cos(d*x+c))^(1/2)/sec(d*x+c)^(3/2)-1/16*(11*A-35*B+63*C)*sin(d*x+ c)/a^2/d/(a+a*cos(d*x+c))^(1/2)/sec(d*x+c)^(1/2)
Time = 10.03 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.77 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\cos ^5\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\left ((-43 A+115 B-219 C) \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+4 \sqrt {2} (8 A-20 B+39 C) \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {1+\sec (c+d x)}}{\sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )}}+\frac {1}{4} (11 A-43 B+73 C+(15 A-55 B+89 C) \cos (c+d x)+(-8 B+10 C) \cos (2 (c+d x))-2 C \cos (3 (c+d x))) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \left (\sin \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {3}{2} (c+d x)\right )\right )\right )}{4 d (a (1+\cos (c+d x)))^{5/2}} \] Input:
Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5 /2)*Sec[c + d*x]^(5/2)),x]
Output:
(Cos[(c + d*x)/2]^5*((((-43*A + 115*B - 219*C)*ArcSin[Tan[(c + d*x)/2]] + 4*Sqrt[2]*(8*A - 20*B + 39*C)*ArcTan[Tan[(c + d*x)/2]/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]])*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2] *Sqrt[1 + Sec[c + d*x]])/Sqrt[Sec[(c + d*x)/2]^2] + ((11*A - 43*B + 73*C + (15*A - 55*B + 89*C)*Cos[c + d*x] + (-8*B + 10*C)*Cos[2*(c + d*x)] - 2*C* Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/4))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))
Time = 2.41 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.99, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 4709, 3042, 3520, 27, 3042, 3456, 27, 3042, 3462, 27, 3042, 3462, 25, 3042, 3461, 3042, 3253, 223, 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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos (c+d x)^2}{\sec (c+d x)^{5/2} (a \cos (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{(\cos (c+d x) a+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) (a (A+7 B-7 C)+4 a (A-B+3 C) \cos (c+d x))}{2 (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) (a (A+7 B-7 C)+4 a (A-B+3 C) \cos (c+d x))}{(\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a (A+7 B-7 C)+4 a (A-B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int -\frac {\cos ^{\frac {3}{2}}(c+d x) \left (5 a^2 (3 A-11 B+19 C)-4 a^2 (7 A-15 B+31 C) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (5 a^2 (3 A-11 B+19 C)-4 a^2 (7 A-15 B+31 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 a^2 (3 A-11 B+19 C)-4 a^2 (7 A-15 B+31 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\int -\frac {2 \sqrt {\cos (c+d x)} \left (3 a^3 (7 A-15 B+31 C)-2 a^3 (11 A-35 B+63 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{2 a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (3 a^3 (7 A-15 B+31 C)-2 a^3 (11 A-35 B+63 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 a^3 (7 A-15 B+31 C)-2 a^3 (11 A-35 B+63 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {\frac {\int -\frac {a^4 (11 A-35 B+63 C)-4 a^4 (8 A-20 B+39 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {\int \frac {a^4 (11 A-35 B+63 C)-4 a^4 (8 A-20 B+39 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {\int \frac {a^4 (11 A-35 B+63 C)-4 a^4 (8 A-20 B+39 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3461 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A-115 B+219 C) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-4 a^3 (8 A-20 B+39 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A-115 B+219 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-4 a^3 (8 A-20 B+39 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A-115 B+219 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx+\frac {8 a^3 (8 A-20 B+39 C) \int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A-115 B+219 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {8 a^{7/2} (8 A-20 B+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {-\frac {2 a^5 (43 A-115 B+219 C) \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}-\frac {8 a^{7/2} (8 A-20 B+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {2 a^2 (7 A-15 B+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {-\frac {\frac {\sqrt {2} a^{7/2} (43 A-115 B+219 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {8 a^{7/2} (8 A-20 B+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A-35 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}}{4 a^2}-\frac {a (3 A-11 B+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
Input:
Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Se c[c + d*x]^(5/2)),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/4*((A - B + C)*Cos[c + d*x]^(7/2 )*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(5/2)) + (-1/2*(a*(3*A - 11*B + 19 *C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(3/2)) - ((-2 *a^2*(7*A - 15*B + 31*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Co s[c + d*x]]) - (-(((-8*a^(7/2)*(8*A - 20*B + 39*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (Sqrt[2]*a^(7/2)*(43*A - 115*B + 219 *C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*C os[c + d*x]])])/d)/a) - (2*a^3*(11*A - 35*B + 63*C)*Sqrt[Cos[c + d*x]]*Sin [c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]))/a)/(4*a^2))/(8*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_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-2/f Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Co s[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && E qQ[a^2 - b^2, 0] && EqQ[d, 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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, 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] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[(A*b - a*B)/b Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) , x], x] + Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]] , x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 - b^2, 0] && NeQ[c^2 - d^2, 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)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Sin[e + f*x])^m*(c + d*S in[e + f*x])^(n - 1)*Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(a*A - b*B + a*C)*Cos[e + f*x]*(a + b* Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x ] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a *d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c *(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c ^2 - d^2, 0] && LtQ[m, -2^(-1)]
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Time = 4.82 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {\left (\left (32 \cos \left (d x +c \right )^{2}+64 \cos \left (d x +c \right )+32\right ) \sqrt {2}\, A \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )+\left (-80 \cos \left (d x +c \right )^{2}-160 \cos \left (d x +c \right )-80\right ) \sqrt {2}\, B \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )+\left (156 \cos \left (d x +c \right )^{2}+312 \cos \left (d x +c \right )+156\right ) \sqrt {2}\, C \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )+\sin \left (d x +c \right ) \left (-15 \cos \left (d x +c \right )-11\right ) \sqrt {2}\, A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\sin \left (d x +c \right ) \left (16 \cos \left (d x +c \right )^{2}+55 \cos \left (d x +c \right )+35\right ) \sqrt {2}\, B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\sin \left (d x +c \right ) \left (8 \cos \left (d x +c \right )^{3}-20 \cos \left (d x +c \right )^{2}-95 \cos \left (d x +c \right )-63\right ) \sqrt {2}\, C \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (43 \cos \left (d x +c \right )^{2}+86 \cos \left (d x +c \right )+43\right ) A \arcsin \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )+\left (-115 \cos \left (d x +c \right )^{2}-230 \cos \left (d x +c \right )-115\right ) B \arcsin \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )+\left (219 \cos \left (d x +c \right )^{2}+438 \cos \left (d x +c \right )+219\right ) C \arcsin \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )\right ) \sqrt {\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}}{16 d \,a^{3} \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 ) \sqrt {\sec \left (d x +c \right )}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(502\) |
| parts | \(\frac {A \sqrt {2}\, \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sqrt {2}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \left (32+64 \sec \left (d x +c \right )+32 \sec \left (d x +c \right )^{2}\right )+\sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (-15 \tan \left (d x +c \right )-11 \tan \left (d x +c \right ) \sec \left (d x +c \right )\right )+\arcsin \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \left (43+86 \sec \left (d x +c \right )+43 \sec \left (d x +c \right )^{2}\right )\right )}{32 d \,a^{3} \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 ) \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}+\frac {B \sqrt {2}\, \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sqrt {2}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \left (-80-160 \sec \left (d x +c \right )-80 \sec \left (d x +c \right )^{2}\right )+\sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (16 \sin \left (d x +c \right )+55 \tan \left (d x +c \right )+35 \tan \left (d x +c \right ) \sec \left (d x +c \right )\right )+\arcsin \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \left (-115-230 \sec \left (d x +c \right )-115 \sec \left (d x +c \right )^{2}\right )\right )}{32 d \,a^{3} \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 ) \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}+\frac {C \sqrt {\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\, \left (\sqrt {2}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \left (156+312 \sec \left (d x +c \right )+156 \sec \left (d x +c \right )^{2}\right )+\left (8 \cos \left (d x +c \right )^{3}-20 \cos \left (d x +c \right )^{2}-95 \cos \left (d x +c \right )-63\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )+\arcsin \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \left (219+438 \sec \left (d x +c \right )+219 \sec \left (d x +c \right )^{2}\right )\right )}{16 d \,a^{3} \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 ) \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(677\) |
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c)^(5/2 ),x,method=_RETURNVERBOSE)
Output:
1/16/d/a^3*((32*cos(d*x+c)^2+64*cos(d*x+c)+32)*2^(1/2)*A*arctan((cos(d*x+c )/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))+(-80*cos(d*x+c)^2-160*cos(d*x+c)-80)*2 ^(1/2)*B*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))+(156*cos(d*x +c)^2+312*cos(d*x+c)+156)*2^(1/2)*C*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/ 2)*tan(d*x+c))+sin(d*x+c)*(-15*cos(d*x+c)-11)*2^(1/2)*A*(cos(d*x+c)/(1+cos (d*x+c)))^(1/2)+sin(d*x+c)*(16*cos(d*x+c)^2+55*cos(d*x+c)+35)*2^(1/2)*B*(c os(d*x+c)/(1+cos(d*x+c)))^(1/2)+sin(d*x+c)*(8*cos(d*x+c)^3-20*cos(d*x+c)^2 -95*cos(d*x+c)-63)*2^(1/2)*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+(43*cos(d*x +c)^2+86*cos(d*x+c)+43)*A*arcsin(-csc(d*x+c)+cot(d*x+c))+(-115*cos(d*x+c)^ 2-230*cos(d*x+c)-115)*B*arcsin(-csc(d*x+c)+cot(d*x+c))+(219*cos(d*x+c)^2+4 38*cos(d*x+c)+219)*C*arcsin(-csc(d*x+c)+cot(d*x+c)))*(cos(1/2*d*x+1/2*c)^2 *a)^(1/2)/(cos(d*x+c)^3+3*cos(d*x+c)^2+3*cos(d*x+c)+1)/sec(d*x+c)^(1/2)/(c os(d*x+c)/(1+cos(d*x+c)))^(1/2)
Time = 79.06 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.02 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left ({\left (43 \, A - 115 \, B + 219 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (43 \, A - 115 \, B + 219 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (43 \, A - 115 \, B + 219 \, C\right )} \cos \left (d x + c\right ) + 43 \, A - 115 \, B + 219 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 8 \, {\left ({\left (8 \, A - 20 \, B + 39 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A - 20 \, B + 39 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, A - 20 \, B + 39 \, C\right )} \cos \left (d x + c\right ) + 8 \, A - 20 \, B + 39 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left (8 \, C \cos \left (d x + c\right )^{4} + 4 \, {\left (4 \, B - 5 \, C\right )} \cos \left (d x + c\right )^{3} - 5 \, {\left (3 \, A - 11 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (11 \, A - 35 \, B + 63 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{32 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c )^(5/2),x, algorithm="fricas")
Output:
1/32*(sqrt(2)*((43*A - 115*B + 219*C)*cos(d*x + c)^3 + 3*(43*A - 115*B + 2 19*C)*cos(d*x + c)^2 + 3*(43*A - 115*B + 219*C)*cos(d*x + c) + 43*A - 115* B + 219*C)*sqrt(a)*arctan(sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - 8*((8*A - 20*B + 39*C)*cos(d*x + c)^3 + 3*(8 *A - 20*B + 39*C)*cos(d*x + c)^2 + 3*(8*A - 20*B + 39*C)*cos(d*x + c) + 8* A - 20*B + 39*C)*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c) )/(sqrt(a)*sin(d*x + c))) + 2*(8*C*cos(d*x + c)^4 + 4*(4*B - 5*C)*cos(d*x + c)^3 - 5*(3*A - 11*B + 19*C)*cos(d*x + c)^2 - (11*A - 35*B + 63*C)*cos(d *x + c))*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^3*d* cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**(5/2)/sec(d*x +c)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c )^(5/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((a*cos(d*x + c) + a)^(5 /2)*sec(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c )^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(5/2)*(a + a *cos(c + d*x))^(5/2)),x)
Output:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(5/2)*(a + a *cos(c + d*x))^(5/2)), x)
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}+\sec \left (d x +c \right )^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}+\sec \left (d x +c \right )^{3}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}+\sec \left (d x +c \right )^{3}}d x \right ) a \right )}{a^{3}} \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c)^(5/2 ),x)
Output:
(sqrt(a)*(int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1)*cos(c + d*x))/(co s(c + d*x)**3*sec(c + d*x)**3 + 3*cos(c + d*x)**2*sec(c + d*x)**3 + 3*cos( c + d*x)*sec(c + d*x)**3 + sec(c + d*x)**3),x)*b + int((sqrt(sec(c + d*x)) *sqrt(cos(c + d*x) + 1)*cos(c + d*x)**2)/(cos(c + d*x)**3*sec(c + d*x)**3 + 3*cos(c + d*x)**2*sec(c + d*x)**3 + 3*cos(c + d*x)*sec(c + d*x)**3 + sec (c + d*x)**3),x)*c + int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1))/(cos( c + d*x)**3*sec(c + d*x)**3 + 3*cos(c + d*x)**2*sec(c + d*x)**3 + 3*cos(c + d*x)*sec(c + d*x)**3 + sec(c + d*x)**3),x)*a))/a**3