Integrand size = 43, antiderivative size = 267 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {4 a^3 (21 A+17 B+15 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (11 B+6 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (99 A+143 B+105 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d} \]
4/15*a^3*(21*A+17*B+15*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+4/231*a^3*(143*A+121*B+105*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+4/1155*a^3*(264*A+253*B+210*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/1 1*C*cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+2/99*(11*B+6*C)*cos(d *x+c)^(3/2)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d+2/693*(99*A+143*B+105*C) *cos(d*x+c)^(3/2)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d+4/231*a^3*(143*A+121*B +105*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.39 (sec) , antiderivative size = 1374, normalized size of antiderivative = 5.15 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((21 *A + 17*B + 15*C)*Cot[c])/d + ((2354*A + 2134*B + 1953*C)*Cos[d*x]*Sin[c]) /(7392*d) + ((54*A + 73*B + 75*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((44*A + 132*B + 189*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((B + 3*C)*Cos[4*d*x]*Sin[4 *c])/(288*d) + (C*Cos[5*d*x]*Sin[5*c])/(704*d) + ((2354*A + 2134*B + 1953* C)*Cos[c]*Sin[d*x])/(7392*d) + ((54*A + 73*B + 75*C)*Cos[2*c]*Sin[2*d*x])/ (720*d) + ((44*A + 132*B + 189*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((B + 3* C)*Cos[4*c]*Sin[4*d*x])/(288*d) + (C*Cos[5*c]*Sin[5*d*x])/(704*d)) - (13*A *(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d* x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt [1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot [c]^2]) - (11*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2} , {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTa n[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*S in[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42 *d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*Hypergeometric PFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*S ec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcT...
Time = 1.83 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 27, 3042, 3447, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 (a (11 A+3 C)+a (11 B+6 C) \cos (c+d x))dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 (a (11 A+3 C)+a (11 B+6 C) \cos (c+d x))dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (11 A+3 C)+a (11 B+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (33 A+11 B+15 C) a^2+(99 A+143 B+105 C) \cos (c+d x) a^2\right )dx+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (33 A+11 B+15 C) a^2+(99 A+143 B+105 C) \cos (c+d x) a^2\right )dx+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (33 A+11 B+15 C) a^2+(99 A+143 B+105 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int 3 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (33 A+22 B+21 C) a^3+(264 A+253 B+210 C) \cos (c+d x) a^3\right )dx+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (33 A+22 B+21 C) a^3+(264 A+253 B+210 C) \cos (c+d x) a^3\right )dx+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (33 A+22 B+21 C) a^3+(264 A+253 B+210 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\cos (c+d x)} \left ((264 A+253 B+210 C) \cos ^2(c+d x) a^4+5 (33 A+22 B+21 C) a^4+\left (5 (33 A+22 B+21 C) a^4+(264 A+253 B+210 C) a^4\right ) \cos (c+d x)\right )dx+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left ((264 A+253 B+210 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+5 (33 A+22 B+21 C) a^4+\left (5 (33 A+22 B+21 C) a^4+(264 A+253 B+210 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (77 (21 A+17 B+15 C) a^4+15 (143 A+121 B+105 C) \cos (c+d x) a^4\right )dx+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \sqrt {\cos (c+d x)} \left (77 (21 A+17 B+15 C) a^4+15 (143 A+121 B+105 C) \cos (c+d x) a^4\right )dx+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 (21 A+17 B+15 C) a^4+15 (143 A+121 B+105 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (143 A+121 B+105 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^4 (21 A+17 B+15 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (21 A+17 B+15 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 a^4 (143 A+121 B+105 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (21 A+17 B+15 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 a^4 (143 A+121 B+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (21 A+17 B+15 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 a^4 (143 A+121 B+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (143 A+121 B+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 a^4 (21 A+17 B+15 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}+\frac {6}{7} \left (\frac {2 a^4 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {154 a^4 (21 A+17 B+15 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+15 a^4 (143 A+121 B+105 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {2 (11 B+6 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\) |
(2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + ((2* (11*B + 6*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/( 9*d) + ((2*(99*A + 143*B + 105*C)*Cos[c + d*x]^(3/2)*(a^4 + a^4*Cos[c + d* x])*Sin[c + d*x])/(7*d) + (6*((2*a^4*(264*A + 253*B + 210*C)*Cos[c + d*x]^ (3/2)*Sin[c + d*x])/(5*d) + ((154*a^4*(21*A + 17*B + 15*C)*EllipticE[(c + d*x)/2, 2])/d + 15*a^4*(143*A + 121*B + 105*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5))/7)/9)/(11*a)
3.5.48.3.1 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[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !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)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 25.59 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.04
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (10080 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6160 B -43680 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (3960 A +24200 B +77280 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-14256 A -37532 B -72240 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (19866 A +29722 B +39270 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-6864 A -8118 B -8820 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+2145 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-4851 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1815 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3927 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1575 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3465 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(545\) |
| parts | \(\text {Expression too large to display}\) | \(1214\) |
int((a+cos(d*x+c)*a)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, method=_RETURNVERBOSE)
-4/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(10080 *C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-6160*B-43680*C)*sin(1/2*d*x+ 1/2*c)^10*cos(1/2*d*x+1/2*c)+(3960*A+24200*B+77280*C)*sin(1/2*d*x+1/2*c)^8 *cos(1/2*d*x+1/2*c)+(-14256*A-37532*B-72240*C)*sin(1/2*d*x+1/2*c)^6*cos(1/ 2*d*x+1/2*c)+(19866*A+29722*B+39270*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)+(-6864*A-8118*B-8820*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+2145* A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF( cos(1/2*d*x+1/2*c),2^(1/2))-4851*A*(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))+1815*B*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2* d*x+1/2*c),2^(1/2))-3927*B*(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))+1575*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)*EllipticF(cos(1/2*d*x+1/2* c),2^(1/2))-3465*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*sin(1/2*d*x+1/2*c)^4+sin (1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/ 2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.99 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (315 \, C a^{3} \cos \left (d x + c\right )^{4} + 385 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 33 \, B + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 77 \, {\left (27 \, A + 34 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right ) + 30 \, {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d} \]
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1 /2),x, algorithm="fricas")
-2/3465*(15*I*sqrt(2)*(143*A + 121*B + 105*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(143*A + 121*B + 105*C)*a ^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt( 2)*(21*A + 17*B + 15*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(2)*(21*A + 17*B + 15*C)*a ^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c))) - (315*C*a^3*cos(d*x + c)^4 + 385*(B + 3*C)*a^3*cos(d*x + c)^3 + 45*(11*A + 33*B + 42*C)*a^3*cos(d*x + c)^2 + 77*(27*A + 34*B + 30*C)*a^3 *cos(d*x + c) + 30*(143*A + 121*B + 105*C)*a^3)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \]
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1 /2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3*s qrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \]
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1 /2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3*s qrt(cos(d*x + c)), x)
Time = 2.97 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.90 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2\,\left (A\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {B\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {6\,A\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,B\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*(A*a^3*ellipticE(c/2 + (d*x)/2, 2) + A*a^3*ellipticF(c/2 + (d*x)/2, 2) + A*a^3*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (B*a^3*((2*cos(c + d*x)^(1/2 )*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (6*A*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7 *d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hype rgeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (6 *B*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2) ^(1/2)) - (2*B*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d *x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*( sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeo m([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (6*C*a ^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d *x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d*x)^ 2)^(1/2))