Integrand size = 37, antiderivative size = 544 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^4+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (12 a A b^2+16 A b^3+6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (8 A b^2+a^2 (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 A b \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \] Output:
-2/315*(a-b)*(a+b)^(1/2)*(16*A*b^4+6*a^2*b^2*(4*A+7*C)-21*a^4*(7*A+9*C))*c os(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/co s(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+ sec(d*x+c))/(a-b))^(1/2)/a^5/d/sec(d*x+c)^(1/2)-2/315*(a-b)*(a+b)^(1/2)*(1 2*a*A*b^2+16*A*b^3+6*a^2*b*(6*A+7*C)+21*a^3*(7*A+9*C))*cos(d*x+c)^(1/2)*cs c(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-( a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b)) ^(1/2)/a^4/d/sec(d*x+c)^(1/2)+2/315*b*(8*A*b^2+a^2*(13*A+21*C))*(a+b*cos(d *x+c))^(1/2)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d-2/315*(6*A*b^2-7*a^2*(7*A+9 *C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(5/2)*sin(d*x+c)/a^2/d+2/63*A*b*(a+ b*cos(d*x+c))^(1/2)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d+2/9*A*(a+b*cos(d*x+c)) ^(1/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d
Time = 18.96 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.09 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \left (-2 (a+b) \left (-16 A b^4-6 a^2 b^2 (4 A+7 C)+21 a^4 (7 A+9 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )+2 a (a+b) \left (12 a A b^2-16 A b^3-6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+\left (16 A b^4+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \cos (c+d x) (a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 a^4 d \sqrt {a+b \cos (c+d x)} \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )}}+\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \left (147 a^4 A-24 a^2 A b^2-16 A b^4+189 a^4 C-42 a^2 b^2 C\right ) \sin (c+d x)}{315 a^4}+\frac {2 \sec ^2(c+d x) \left (49 a^2 A \sin (c+d x)-6 A b^2 \sin (c+d x)+63 a^2 C \sin (c+d x)\right )}{315 a^2}+\frac {2 \sec (c+d x) \left (13 a^2 A b \sin (c+d x)+8 A b^3 \sin (c+d x)+21 a^2 b C \sin (c+d x)\right )}{315 a^3}+\frac {2 A b \sec ^2(c+d x) \tan (c+d x)}{63 a}+\frac {2}{9} A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11 /2),x]
Output:
(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-16*A*b^4 - 6*a^2*b^ 2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]* Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[T an[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[A rcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x) /2]^2*Tan[(c + d*x)/2]))/(315*a^4*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 189*a^4*C - 42*a^2*b^2*C)*Sin[c + d*x])/(315* a^4) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] - 6*A*b^2*Sin[c + d*x] + 6 3*a^2*C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(13*a^2*A*b*Sin[c + d*x ] + 8*A*b^3*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/(315*a^3) + (2*A*b*Se c[c + d*x]^2*Tan[c + d*x])/(63*a) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/ d
Time = 2.96 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 4709, 3042, 3527, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \sec ^{\frac {11}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{11/2} \sqrt {a+b \cos (c+d x)} \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {a+b \cos (c+d x)} \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {3 b (2 A+3 C) \cos ^2(c+d x)+a (7 A+9 C) \cos (c+d x)+A b}{2 \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 A+3 C) \cos ^2(c+d x)+a (7 A+9 C) \cos (c+d x)+A b}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (7 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+A b}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 \int -\frac {-7 (7 A+9 C) a^2-b (47 A+63 C) \cos (c+d x) a+6 A b^2-4 A b^2 \cos ^2(c+d x)}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-7 (7 A+9 C) a^2-b (47 A+63 C) \cos (c+d x) a+6 A b^2-4 A b^2 \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-7 (7 A+9 C) a^2-b (47 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2-4 A b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-2 b \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)+a \left (21 (7 A+9 C) a^2+2 A b^2\right ) \cos (c+d x)+3 b \left ((13 A+21 C) a^2+8 A b^2\right )}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)+a \left (21 (7 A+9 C) a^2+2 A b^2\right ) \cos (c+d x)+3 b \left ((13 A+21 C) a^2+8 A b^2\right )}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (21 (7 A+9 C) a^2+2 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b \left ((13 A+21 C) a^2+8 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {3 \left (-21 (7 A+9 C) a^4+6 b^2 (4 A+7 C) a^2+b \left (4 A b^2-3 a^2 (37 A+49 C)\right ) \cos (c+d x) a+16 A b^4\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-21 (7 A+9 C) a^4+6 b^2 (4 A+7 C) a^2+b \left (4 A b^2-3 a^2 (37 A+49 C)\right ) \cos (c+d x) a+16 A b^4}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-21 (7 A+9 C) a^4+6 b^2 (4 A+7 C) a^2+b \left (4 A b^2-3 a^2 (37 A+49 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+16 A b^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+(a-b) \left (21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \left (21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
Input:
Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - ((2*(6*A*b^2 - 7*a^2*(7*A + 9*C))*Sq rt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (-(((2*( a - b)*Sqrt[a + b]*(16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C)) *Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[ Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]* Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) + (2*(a - b)*Sqrt[a + b]*(12 *a*A*b^2 + 16*A*b^3 + 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Cot[c + d* x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x ]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d))/a) + (2*b*(8*A*b^2 + a^2*(13*A + 21*C))* Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Cos[c + d*x]^(3/2)))/(5*a))/(7 *a))/9)
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[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*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] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
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[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
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. \(2515\) vs. \(2(486)=972\).
Time = 53.46 (sec) , antiderivative size = 2516, normalized size of antiderivative = 4.62
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2516\) |
| parts | \(\text {Expression too large to display}\) | \(2534\) |
Input:
int((a+b*cos(d*x+c))^(1/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x,method=_ RETURNVERBOSE)
Output:
2/315/d/a^4*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(11/2)/(b*cos(d*x+c)^2+a*cos (d*x+c)+b*cos(d*x+c)+a)*(A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2) *(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(- (a-b)/(a+b))^(1/2))*(147*cos(d*x+c)^7+294*cos(d*x+c)^6+147*cos(d*x+c)^5)+A *(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c) ))^(1/2)*b^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-16*c os(d*x+c)^7-32*cos(d*x+c)^6-16*cos(d*x+c)^5)+C*(1/(a+b)*(a+b*cos(d*x+c))/( 1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticE(-csc( d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(189*cos(d*x+c)^7+378*cos(d*x+c)^6 +189*cos(d*x+c)^5)+A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos( d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/ (a+b))^(1/2))*(-147*cos(d*x+c)^7-294*cos(d*x+c)^6-147*cos(d*x+c)^5)+C*(1/( a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1 /2)*a^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-189*cos(d *x+c)^7-378*cos(d*x+c)^6-189*cos(d*x+c)^5)+sin(d*x+c)*cos(d*x+c)^2*(147*co s(d*x+c)^4+62*cos(d*x+c)^3+62*cos(d*x+c)^2+40*cos(d*x+c)+40)*A*a^4*b+sin(d *x+c)*cos(d*x+c)^3*(13*cos(d*x+c)^3-11*cos(d*x+c)^2-cos(d*x+c)-1)*A*a^3*b^ 2+sin(d*x+c)*cos(d*x+c)^4*(-24*cos(d*x+c)^2+2*cos(d*x+c)+2)*a^2*A*b^3+sin( d*x+c)*cos(d*x+c)^5*(8*cos(d*x+c)-8)*A*a*b^4+sin(d*x+c)*cos(d*x+c)^4*(189* cos(d*x+c)^2+84*cos(d*x+c)+84)*C*a^4*b+sin(d*x+c)*cos(d*x+c)^5*(21*cos(...
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, a lgorithm="fricas")
Output:
integral((C*cos(d*x + c)^2 + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/ 2), x)
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(1/2)*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(11/2),x )
Output:
Timed out
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, a lgorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11 /2), x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, a lgorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11 /2), x)
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1 /2),x)
Output:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1 /2), x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a \] Input:
int((a+b*cos(d*x+c))^(1/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x)
Output:
int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d* x)**5,x)*c + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*sec(c + d*x)* *5,x)*a