Integrand size = 37, antiderivative size = 334 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {(8 A+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{4 a^{5/2} d}-\frac {(43 A+219 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{16 \sqrt {2} a^{5/2} d}-\frac {(A+C) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {(3 A+19 C) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(7 A+31 C) \sin (c+d x)}{16 a^2 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}-\frac {(11 A+63 C) \sin (c+d x)}{16 a^2 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \]
-1/4*(A+C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c)^(7/2)-1/16*(3*A+ 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+31* C)*sin(d*x+c)/a^2/d/sec(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(1/2)-1/16*(11*A+63* C)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(1/2)/sec(d*x+c)^(1/2)+1/4*(8*A+39*C) *arcsin(sin(d*x+c)*a^(1/2)/(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+219*C)*arctan(1/2*sin(d*x+c)*a^(1/2)*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)/a^(5/2)/d*2^(1/2)
Result contains complex when optimal does not.
Time = 8.14 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.90 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {11 i A 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 ) \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{4 d (a (1+\cos (c+d x)))^{5/2}}-\frac {63 i C 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 ) \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{4 d (a (1+\cos (c+d x)))^{5/2}}+\frac {4 i \sqrt {2} A 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)}} \left (-\text {arcsinh}\left (e^{i (c+d x)}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (a (1+\cos (c+d x)))^{5/2}}+\frac {39 i C 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)}} \left (-\text {arcsinh}\left (e^{i (c+d x)}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {2} d (a (1+\cos (c+d x)))^{5/2}}+\frac {\cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (-\frac {3 (5 A+3 C) \cos \left (\frac {d x}{2}\right ) \sin \left (\frac {c}{2}\right )}{2 d}-\frac {10 C \cos \left (\frac {3 d x}{2}\right ) \sin \left (\frac {3 c}{2}\right )}{d}+\frac {C \cos \left (\frac {5 d x}{2}\right ) \sin \left (\frac {5 c}{2}\right )}{d}-\frac {3 (5 A+3 C) \cos \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )}{2 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-A \sin \left (\frac {d x}{2}\right )-C \sin \left (\frac {d x}{2}\right )\right )}{2 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (19 A \sin \left (\frac {d x}{2}\right )+35 C \sin \left (\frac {d x}{2}\right )\right )}{4 d}-\frac {10 C \cos \left (\frac {3 c}{2}\right ) \sin \left (\frac {3 d x}{2}\right )}{d}+\frac {C \cos \left (\frac {5 c}{2}\right ) \sin \left (\frac {5 d x}{2}\right )}{d}+\frac {(19 A+35 C) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{4 d}-\frac {(A+C) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{2 d}\right )}{(a (1+\cos (c+d x)))^{5/2}} \]
(((-11*I)/4)*A*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))])]*Cos[c/2 + (d*x)/2]^5)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) - (((63*I)/4)*C*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))])]*Cos[c/2 + (d*x)/2]^5)/(d*E^((I/2)*(c + d *x))*(a*(1 + Cos[c + d*x]))^(5/2)) + ((4*I)*Sqrt[2]*A*Sqrt[E^(I*(c + d*x)) /(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-ArcSinh[E^(I*( c + d*x))] + Sqrt[2]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^(( 2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d* x)/2]^5)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) + ((39*I)*C* Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x ))]*(-ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*ArcTanh[(-1 + E^(I*(c + d*x)))/(S qrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d* x))]])*Cos[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) + (Cos[c/2 + (d*x)/2]^5*Sqrt[Sec[c + d*x]]*((-3*(5*A + 3*C )*Cos[(d*x)/2]*Sin[c/2])/(2*d) - (10*C*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d + (C *Cos[(5*d*x)/2]*Sin[(5*c)/2])/d - (3*(5*A + 3*C)*Cos[c/2]*Sin[(d*x)/2])/(2 *d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-(A*Sin[(d*x)/2]) - C*Sin[(d*x)/2])) /(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(19*A*Sin[(d*x)/2] + 35*C*Sin[(...
Time = 2.19 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.99, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.568, Rules used = {3042, 4709, 3042, 3521, 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.
| \(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \cos (c+d x)^2}{\sec (c+d x)^{5/2} (a \cos (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (C \cos ^2(c+d x)+A\right )}{(\cos (c+d x) a+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) (a (A-7 C)+4 a (A+3 C) \cos (c+d x))}{2 (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) (a (A-7 C)+4 a (A+3 C) \cos (c+d x))}{(\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a (A-7 C)+4 a (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int -\frac {\cos ^{\frac {3}{2}}(c+d x) \left (5 a^2 (3 A+19 C)-4 a^2 (7 A+31 C) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (5 a^2 (3 A+19 C)-4 a^2 (7 A+31 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 a^2 (3 A+19 C)-4 a^2 (7 A+31 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\int -\frac {2 \sqrt {\cos (c+d x)} \left (3 a^3 (7 A+31 C)-2 a^3 (11 A+63 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{2 a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (3 a^3 (7 A+31 C)-2 a^3 (11 A+63 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 a^3 (7 A+31 C)-2 a^3 (11 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {\frac {\int -\frac {a^4 (11 A+63 C)-4 a^4 (8 A+39 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {\int \frac {a^4 (11 A+63 C)-4 a^4 (8 A+39 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {\int \frac {a^4 (11 A+63 C)-4 a^4 (8 A+39 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3461 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A+219 C) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-4 a^3 (8 A+39 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A+219 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-4 a^3 (8 A+39 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A+219 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx+\frac {8 a^3 (8 A+39 C) \int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {a^4 (43 A+219 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {8 a^{7/2} (8 A+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {-\frac {-\frac {2 a^5 (43 A+219 C) \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}-\frac {8 a^{7/2} (8 A+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {-\frac {2 a^2 (7 A+31 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {-\frac {\frac {\sqrt {2} a^{7/2} (43 A+219 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {8 a^{7/2} (8 A+39 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {2 a^3 (11 A+63 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{a}}{4 a^2}-\frac {a (3 A+19 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/4*((A + C)*Cos[c + d*x]^(7/2)*Si n[c + d*x])/(d*(a + a*Cos[c + d*x])^(5/2)) + (-1/2*(a*(3*A + 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 + 31*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) - (-( ((-8*a^(7/2)*(8*A + 39*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (Sqrt[2]*a^(7/2)*(43*A + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x]) /(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d)/a) - (2*a^3*(1 1*A + 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))
3.13.52.3.1 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[((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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[a*(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)) I nt[(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)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(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, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(633\) vs. \(2(281)=562\).
Time = 3.30 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (8 C \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )-20 C \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+32 A \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 )-15 A \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )+156 C \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 )-95 C \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )+64 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 ) \sqrt {2}\, \sec \left (d x +c \right )-11 A \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 )+43 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+312 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 ) \sqrt {2}\, \sec \left (d x +c \right )-63 C \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 )+219 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+32 A \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 (\sec ^{2}\left (d x +c \right )\right )+86 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sec \left (d x +c \right )+156 C \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 (\sec ^{2}\left (d x +c \right )\right )+438 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sec \left (d x +c \right )+43 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )+219 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )\right )}{32 a^{3} d \left (1+\cos \left (d x +c \right )\right )^{3} \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(634\) |
| parts | \(\frac {A \sqrt {2}\, \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (32 \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 )-15 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )+64 \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 ) \sec \left (d x +c \right )-11 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )+43 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+32 \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 (\sec ^{2}\left (d x +c \right )\right )+86 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sec \left (d x +c \right )+43 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )\right )}{32 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{3} \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 {2}\, \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (8 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-20 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+156 \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 )-95 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )+219 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+312 \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 ) \sec \left (d x +c \right )-63 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )+438 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sec \left (d x +c \right )+156 \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 (\sec ^{2}\left (d x +c \right )\right )+219 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )\right )}{32 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{3} \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(680\) |
1/32/a^3/d*2^(1/2)*((1+cos(d*x+c))*a)^(1/2)/(1+cos(d*x+c))^3/sec(d*x+c)^(5 /2)/(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(8*C*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+ c)))^(1/2)*sin(d*x+c)*cos(d*x+c)-20*C*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^ (1/2)*sin(d*x+c)+32*A*2^(1/2)*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan (d*x+c))-15*A*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c)+156*C*2 ^(1/2)*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))-95*C*2^(1/2)*( cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c)+64*A*arctan((cos(d*x+c)/(1+cos (d*x+c)))^(1/2)*tan(d*x+c))*2^(1/2)*sec(d*x+c)-11*A*2^(1/2)*(cos(d*x+c)/(1 +cos(d*x+c)))^(1/2)*tan(d*x+c)*sec(d*x+c)+43*A*arcsin(cot(d*x+c)-csc(d*x+c ))+312*C*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))*2^(1/2)*sec( d*x+c)-63*C*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c)*sec(d*x+c )+219*C*arcsin(cot(d*x+c)-csc(d*x+c))+32*A*2^(1/2)*arctan((cos(d*x+c)/(1+c os(d*x+c)))^(1/2)*tan(d*x+c))*sec(d*x+c)^2+86*A*arcsin(cot(d*x+c)-csc(d*x+ c))*sec(d*x+c)+156*C*2^(1/2)*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan( d*x+c))*sec(d*x+c)^2+438*C*arcsin(cot(d*x+c)-csc(d*x+c))*sec(d*x+c)+43*A*a rcsin(cot(d*x+c)-csc(d*x+c))*sec(d*x+c)^2+219*C*arcsin(cot(d*x+c)-csc(d*x+ c))*sec(d*x+c)^2)
Time = 10.83 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.97 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left ({\left (43 \, A + 219 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (43 \, A + 219 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (43 \, A + 219 \, C\right )} \cos \left (d x + c\right ) + 43 \, A + 219 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 8 \, {\left ({\left (8 \, A + 39 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 39 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, A + 39 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 39 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left (8 \, C \cos \left (d x + c\right )^{4} - 20 \, C \cos \left (d x + c\right )^{3} - 5 \, {\left (3 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (11 \, A + 63 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{32 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
1/32*(sqrt(2)*((43*A + 219*C)*cos(d*x + c)^3 + 3*(43*A + 219*C)*cos(d*x + c)^2 + 3*(43*A + 219*C)*cos(d*x + c) + 43*A + 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 + 39*C)*cos(d*x + c)^3 + 3*(8*A + 39*C)*cos(d*x + c)^2 + 3*(8*A + 39 *C)*cos(d*x + c) + 8*A + 39*C)*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a)*sqr t(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) + 2*(8*C*cos(d*x + c)^4 - 20*C*cos (d*x + c)^3 - 5*(3*A + 19*C)*cos(d*x + c)^2 - (11*A + 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. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]