Integrand size = 43, antiderivative size = 357 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 (2 A b+3 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
-2/15*(27*B*a^2*b+15*B*b^3+9*a*b^2*(3*A+5*C)+a^3*(7*A+9*C))*(cos(1/2*d*x+1 /2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+ 2/21*(5*B*a^3+21*B*a*b^2+7*b^3*(A+3*C)+3*a^2*b*(5*A+7*C))*(cos(1/2*d*x+1/2 *c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/ 315*a*(24*A*b^2+99*B*a*b+7*a^2*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/ 63*(8*A*b^3+15*B*a^3+54*B*a*b^2+9*a^2*b*(5*A+7*C))*sin(d*x+c)/d/cos(d*x+c) ^(3/2)+2/21*(2*A*b+3*B*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(7/2) +2/9*A*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/15*(27*B*a^2*b+1 5*B*b^3+9*a*b^2*(3*A+5*C)+a^3*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)
Time = 11.03 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (-49 a^3 A-189 a A b^2-189 a^2 b B-105 b^3 B-63 a^3 C-315 a b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (75 a^2 A b+35 A b^3+25 a^3 B+105 a b^2 B+105 a^2 b C+105 b^3 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{105 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2}{7} \sec ^4(c+d x) \left (3 a^2 A b \sin (c+d x)+a^3 B \sin (c+d x)\right )+\frac {2}{45} \sec ^3(c+d x) \left (7 a^3 A \sin (c+d x)+27 a A b^2 \sin (c+d x)+27 a^2 b B \sin (c+d x)+9 a^3 C \sin (c+d x)\right )+\frac {2}{21} \sec ^2(c+d x) \left (15 a^2 A b \sin (c+d x)+7 A b^3 \sin (c+d x)+5 a^3 B \sin (c+d x)+21 a b^2 B \sin (c+d x)+21 a^2 b C \sin (c+d x)\right )+\frac {2}{15} \sec (c+d x) \left (7 a^3 A \sin (c+d x)+27 a A b^2 \sin (c+d x)+27 a^2 b B \sin (c+d x)+15 b^3 B \sin (c+d x)+9 a^3 C \sin (c+d x)+45 a b^2 C \sin (c+d x)\right )+\frac {2}{9} a^3 A \sec ^4(c+d x) \tan (c+d x)\right )}{d} \]
Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(11/2),x]
(2*(-49*a^3*A - 189*a*A*b^2 - 189*a^2*b*B - 105*b^3*B - 63*a^3*C - 315*a*b ^2*C)*EllipticE[(c + d*x)/2, 2] + 2*(75*a^2*A*b + 35*A*b^3 + 25*a^3*B + 10 5*a*b^2*B + 105*a^2*b*C + 105*b^3*C)*EllipticF[(c + d*x)/2, 2])/(105*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]^4*(3*a^2*A*b*Sin[c + d*x] + a^3*B*Sin [c + d*x]))/7 + (2*Sec[c + d*x]^3*(7*a^3*A*Sin[c + d*x] + 27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 9*a^3*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]^2*(15*a^2*A*b*Sin[c + d*x] + 7*A*b^3*Sin[c + d*x] + 5*a^3*B*Sin[c + d*x] + 21*a*b^2*B*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/21 + (2*Sec[c + d*x]*(7*a^3*A*Sin[c + d*x] + 27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 15*b^3*B*Sin[c + d*x] + 9*a^3*C*Sin[c + d*x] + 45*a*b^2*C*Sin[c + d*x]))/15 + (2*a^3*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d
Time = 1.99 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.93, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 3116, 3042, 3119, 3120}
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.
\(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (A+9 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+3 (2 A b+3 a B)\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (A+9 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+3 (2 A b+3 a B)\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 (2 A b+3 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (7 (7 A+9 C) a^2+99 b B a+24 A b^2+b (13 A b+63 C b+9 a B) \cos ^2(c+d x)+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (7 (7 A+9 C) a^2+99 b B a+24 A b^2+b (13 A b+63 C b+9 a B) \cos ^2(c+d x)+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (7 (7 A+9 C) a^2+99 b B a+24 A b^2+b (13 A b+63 C b+9 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3510 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 b^2 (13 A b+63 C b+9 a B) \cos ^2(c+d x)+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \cos (c+d x)+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 (13 A b+63 C b+9 a B) \cos ^2(c+d x)+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \cos (c+d x)+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 (13 A b+63 C b+9 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3500 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (7 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right )+5 \left (5 B a^3+3 b (5 A+7 C) a^2+21 b^2 B a+7 b^3 (A+3 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {7 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right )+5 \left (5 B a^3+3 b (5 A+7 C) a^2+21 b^2 B a+7 b^3 (A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {7 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right )+5 \left (5 B a^3+3 b (5 A+7 C) a^2+21 b^2 B a+7 b^3 (A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right )}{d}+7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
(2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*( 2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2 )) + ((2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(5*d*Co s[c + d*x]^(5/2)) + ((10*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + 3*((10*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d + 7 *(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*((-2*Elli pticE[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7 )/9
3.11.86.3.1 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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -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[(-(c^2*C - B*c*d + 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* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(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, B, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1264\) vs. \(2(385)=770\).
Time = 8.99 (sec) , antiderivative size = 1265, normalized size of antiderivative = 3.54
method | result | size |
default | \(\text {Expression too large to display}\) | \(1265\) |
parts | \(\text {Expression too large to display}\) | \(1467\) |
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x ,method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C*b^3*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 /2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 2*A*a^3*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/ 2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*si n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^ 3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2 +1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos( 1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^( 1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+ 1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d *x+1/2*c),2^(1/2))))+2*b^2*(B*b+3*C*a)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x +1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1 /2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2 *d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+2*a^2*(3*A*b +B*a)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c )^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/ 2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+5/ 21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*s...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{3} + 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 i \, B a b^{2} + 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{3} - 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b - 21 i \, B a b^{2} - 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{3} + 27 i \, B a^{2} b + 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} + 15 i \, B b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{3} - 27 i \, B a^{2} b - 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} - 15 i \, B b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (21 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 27 \, B a^{2} b + 9 \, {\left (3 \, A + 5 \, C\right )} a b^{2} + 15 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} + 35 \, A a^{3} + 15 \, {\left (5 \, B a^{3} + 3 \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 \, B a b^{2} + 7 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 27 \, B a^{2} b + 27 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \]
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 1/2),x, algorithm="fricas")
-1/315*(15*sqrt(2)*(5*I*B*a^3 + 3*I*(5*A + 7*C)*a^2*b + 21*I*B*a*b^2 + 7*I *(A + 3*C)*b^3)*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I *sin(d*x + c)) + 15*sqrt(2)*(-5*I*B*a^3 - 3*I*(5*A + 7*C)*a^2*b - 21*I*B*a *b^2 - 7*I*(A + 3*C)*b^3)*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d* x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(I*(7*A + 9*C)*a^3 + 27*I*B*a^2*b + 9*I*(3*A + 5*C)*a*b^2 + 15*I*B*b^3)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(- I*(7*A + 9*C)*a^3 - 27*I*B*a^2*b - 9*I*(3*A + 5*C)*a*b^2 - 15*I*B*b^3)*cos (d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(21*((7*A + 9*C)*a^3 + 27*B*a^2*b + 9*(3*A + 5*C)* a*b^2 + 15*B*b^3)*cos(d*x + c)^4 + 35*A*a^3 + 15*(5*B*a^3 + 3*(5*A + 7*C)* a^2*b + 21*B*a*b^2 + 7*A*b^3)*cos(d*x + c)^3 + 7*((7*A + 9*C)*a^3 + 27*B*a ^2*b + 27*A*a*b^2)*cos(d*x + c)^2 + 45*(B*a^3 + 3*A*a^2*b)*cos(d*x + c))*s qrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 1/2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/c os(d*x + c)^(11/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 1/2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/c os(d*x + c)^(11/2), x)
Time = 8.69 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {70\,A\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{4},\frac {1}{2};\ -\frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )+210\,A\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+378\,A\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+270\,A\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{315\,d\,{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {\frac {2\,B\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7}+2\,B\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+2\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+\frac {6\,B\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5}}{d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {2\,C\,b^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {6\,C\,a\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^2\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(70*A*a^3*sin(c + d*x)*hypergeom([-9/4, 1/2], -5/4, cos(c + d*x)^2) + 210* A*b^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x) ^2) + 378*A*a*b^2*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2) + 270*A*a^2*b*cos(c + d*x)*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(315*d*cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2 )^(1/2)) + ((2*B*a^3*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x )^2))/7 + 2*B*b^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2) + 2*B*a*b^2*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-3/4, 1 /2], 1/4, cos(c + d*x)^2) + (6*B*a^2*b*cos(c + d*x)*sin(c + d*x)*hypergeom ([-5/4, 1/2], -1/4, cos(c + d*x)^2))/5)/(d*cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (2*C*b^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*C*a^3*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5 /2)*(sin(c + d*x)^2)^(1/2)) + (6*C*a*b^2*sin(c + d*x)*hypergeom([-1/4, 1/2 ], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + ( 2*C*a^2*b*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(d*cos (c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2))