Integrand size = 37, antiderivative size = 544 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^4+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (12 a A b^2+16 A b^3+6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (8 A b^2+a^2 (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 A b \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
2/315*b*(8*A*b^2+a^2*(13*A+21*C))*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*cos(d*x +c))^(1/2)/a^3/d-2/315*(6*A*b^2-7*a^2*(7*A+9*C))*sec(d*x+c)^(5/2)*sin(d*x+ c)*(a+b*cos(d*x+c))^(1/2)/a^2/d+2/63*A*b*sec(d*x+c)^(7/2)*sin(d*x+c)*(a+b* cos(d*x+c))^(1/2)/a/d+2/9*A*sec(d*x+c)^(9/2)*sin(d*x+c)*(a+b*cos(d*x+c))^( 1/2)/d-2/315*(a-b)*(16*A*b^4+6*a^2*b^2*(4*A+7*C)-21*a^4*(7*A+9*C))*csc(d*x +c)*EllipticE((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+b)^(1/2)*cos(d*x+c)^(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)*(12*a*A* b^2+16*A*b^3+6*a^2*b*(6*A+7*C)+21*a^3*(7*A+9*C))*csc(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+b )^(1/2)*cos(d*x+c)^(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)
Leaf count is larger than twice the leaf count of optimal. \(3619\) vs. \(2(544)=1088\).
Time = 22.97 (sec) , antiderivative size = 3619, normalized size of antiderivative = 6.65 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Result too large to show} \]
(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] + 63*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*Sec[c + d*x]^2*Ta n[c + d*x])/(63*a) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d + (2*((-7*a*A )/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*A*b^2)/(105*a*Sqrt [a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4)/(315*a^3*Sqrt[a + b* Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sq rt[Sec[c + d*x]]) + (2*b^2*C)/(15*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b*Sqrt[Sec[c + d*x]])/(35*Sqrt[a + b*Cos[c + d*x]]) + (4*A*b ^3*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Sqrt[ Sec[c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x]]) - (2*b*C*Sqrt[Sec[c + d* x]])/(15*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(15*a^2* Sqrt[a + b*Cos[c + d*x]]) - (7*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(1 5*Sqrt[a + b*Cos[c + d*x]]) + (8*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]] )/(105*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec [c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x]]) - (3*b*C*Cos[2*(c + d*x)]*S qrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Cos[2*(c + d*x) ]*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + ...
Time = 2.84 (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 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 A b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
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)
3.15.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(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. \(5582\) vs. \(2(492)=984\).
Time = 30.00 (sec) , antiderivative size = 5583, normalized size of antiderivative = 10.26
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(5583\) |
| default | \(\text {Expression too large to display}\) | \(5651\) |
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \]