Integrand size = 43, antiderivative size = 303 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {4 a^3 (221 A+195 B+175 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {4 a^3 (121 A+105 B+95 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (221 A+195 B+175 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac {20 a^3 (286 A+273 B+236 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {2 (13 B+6 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+195 B+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d} \] Output:
4/195*a^3*(221*A+195*B+175*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/23 1*a^3*(121*A+105*B+95*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/231*a^ 3*(121*A+105*B+95*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+4/585*a^3*(221*A+195*B+ 175*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+20/9009*a^3*(286*A+273*B+236*C)*cos(d *x+c)^(5/2)*sin(d*x+c)/d+2/13*C*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^3*sin(d* x+c)/d+2/143*(13*B+6*C)*cos(d*x+c)^(5/2)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c) /a/d+2/1287*(143*A+195*B+145*C)*cos(d*x+c)^(5/2)*(a^3+a^3*cos(d*x+c))*sin( d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.57 (sec) , antiderivative size = 1426, normalized size of antiderivative = 4.71 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/390*((2 21*A + 195*B + 175*C)*Cot[c])/d + ((2134*A + 1953*B + 1811*C)*Cos[d*x]*Sin [c])/(7392*d) + ((7592*A + 7800*B + 7825*C)*Cos[2*d*x]*Sin[2*c])/(74880*d) + ((132*A + 189*B + 215*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((13*A + 39*B + 59*C)*Cos[4*d*x]*Sin[4*c])/(3744*d) + ((B + 3*C)*Cos[5*d*x]*Sin[5*c])/(7 04*d) + (C*Cos[6*d*x]*Sin[6*c])/(1664*d) + ((2134*A + 1953*B + 1811*C)*Cos [c]*Sin[d*x])/(7392*d) + ((7592*A + 7800*B + 7825*C)*Cos[2*c]*Sin[2*d*x])/ (74880*d) + ((132*A + 189*B + 215*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((13* A + 39*B + 59*C)*Cos[4*c]*Sin[4*d*x])/(3744*d) + ((B + 3*C)*Cos[5*c]*Sin[5 *d*x])/(704*d) + (C*Cos[6*c]*Sin[6*d*x])/(1664*d)) - (11*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 - A rcTan[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]) - (5*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 - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - A rcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c ]^2]) - (95*C*(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 - ArcT...
Time = 1.90 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 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.
| \(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^3 (a (13 A+5 C)+a (13 B+6 C) \cos (c+d x))dx}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^3 (a (13 A+5 C)+a (13 B+6 C) \cos (c+d x))dx}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (13 A+5 C)+a (13 B+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^2 \left ((143 A+65 B+85 C) a^2+(143 A+195 B+145 C) \cos (c+d x) a^2\right )dx+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^2 \left ((143 A+65 B+85 C) a^2+(143 A+195 B+145 C) \cos (c+d x) a^2\right )dx+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((143 A+65 B+85 C) a^2+(143 A+195 B+145 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a) \left ((1001 A+780 B+745 C) a^3+5 (286 A+273 B+236 C) \cos (c+d x) a^3\right )dx+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((1001 A+780 B+745 C) a^3+5 (286 A+273 B+236 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (5 (286 A+273 B+236 C) \cos ^2(c+d x) a^4+(1001 A+780 B+745 C) a^4+\left (5 (286 A+273 B+236 C) a^4+(1001 A+780 B+745 C) a^4\right ) \cos (c+d x)\right )dx+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 (286 A+273 B+236 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(1001 A+780 B+745 C) a^4+\left (5 (286 A+273 B+236 C) a^4+(1001 A+780 B+745 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {2}{7} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (117 (121 A+105 B+95 C) a^4+77 (221 A+195 B+175 C) \cos (c+d x) a^4\right )dx+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (117 (121 A+105 B+95 C) a^4+77 (221 A+195 B+175 C) \cos (c+d x) a^4\right )dx+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (117 (121 A+105 B+95 C) a^4+77 (221 A+195 B+175 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (121 A+105 B+95 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^4 (221 A+195 B+175 C) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (121 A+105 B+95 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+77 a^4 (221 A+195 B+175 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (221 A+195 B+175 C) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a^4 (121 A+105 B+95 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (221 A+195 B+175 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a^4 (121 A+105 B+95 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (121 A+105 B+95 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 a^4 (221 A+195 B+175 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}+\frac {2}{9} \left (\frac {10 a^4 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {1}{7} \left (77 a^4 (221 A+195 B+175 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a^4 (121 A+105 B+95 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {2 (13 B+6 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
Input:
Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[ c + d*x]^2),x]
Output:
(2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + ((2* (13*B + 6*C)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/( 11*d) + ((2*(143*A + 195*B + 145*C)*Cos[c + d*x]^(5/2)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(9*d) + (2*((10*a^4*(286*A + 273*B + 236*C)*Cos[c + d* x]^(5/2)*Sin[c + d*x])/(7*d) + (117*a^4*(121*A + 105*B + 95*C)*((2*Ellipti cF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)) + 7 7*a^4*(221*A + 195*B + 175*C)*((6*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Co s[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)))/7))/9)/11)/(13*a)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(575\) vs. \(2(278)=556\).
Time = 45.36 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-221760 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (131040 B +1058400 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-80080 A -567840 B -2122400 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (314600 A +1004640 B +2331040 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-487916 A -939120 B -1535860 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (386386 A +510510 B +633710 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-105534 A -114660 B -121230 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+23595 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+20475 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-45045 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+18525 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(576\) |
| parts | \(\text {Expression too large to display}\) | \(1319\) |
Input:
int(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
-4/45045*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-22 1760*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(131040*B+1058400*C)*sin(1 /2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-80080*A-567840*B-2122400*C)*sin(1/2* d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(314600*A+1004640*B+2331040*C)*sin(1/2*d* x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-487916*A-939120*B-1535860*C)*sin(1/2*d*x+1 /2*c)^6*cos(1/2*d*x+1/2*c)+(386386*A+510510*B+633710*C)*sin(1/2*d*x+1/2*c) ^4*cos(1/2*d*x+1/2*c)+(-105534*A-114660*B-121230*C)*sin(1/2*d*x+1/2*c)^2*c os(1/2*d*x+1/2*c)+23595*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))-51051*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))+20475*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))-45045*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))+18525*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))-40425*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)/( -1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.95 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (195 i \, \sqrt {2} {\left (121 \, A + 105 \, B + 95 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {2} {\left (121 \, A + 105 \, B + 95 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (221 \, A + 195 \, B + 175 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (221 \, A + 195 \, B + 175 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3465 \, C a^{3} \cos \left (d x + c\right )^{5} + 4095 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 385 \, {\left (13 \, A + 39 \, B + 50 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 585 \, {\left (33 \, A + 42 \, B + 38 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 154 \, {\left (221 \, A + 195 \, B + 175 \, C\right )} a^{3} \cos \left (d x + c\right ) + 390 \, {\left (121 \, A + 105 \, B + 95 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{45045 \, d} \] Input:
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
Output:
-2/45045*(195*I*sqrt(2)*(121*A + 105*B + 95*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 195*I*sqrt(2)*(121*A + 105*B + 95*C)* a^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt (2)*(221*A + 195*B + 175*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse (-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(2)*(221*A + 195*B + 1 75*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3465*C*a^3*cos(d*x + c)^5 + 4095*(B + 3*C)*a^3*cos(d* x + c)^4 + 385*(13*A + 39*B + 50*C)*a^3*cos(d*x + c)^3 + 585*(33*A + 42*B + 38*C)*a^3*cos(d*x + c)^2 + 154*(221*A + 195*B + 175*C)*a^3*cos(d*x + c) + 390*(121*A + 105*B + 95*C)*a^3)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(3/2)*(a+a*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+ c)**2),x)
Output:
Timed out
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3*c os(d*x + c)^(3/2), x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3*c os(d*x + c)^(3/2), x)
Time = 1.22 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.80 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos( c + d*x)^2),x)
Output:
(A*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)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d* (sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hyperg eom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2 *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)) - (6*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*B*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)) - (2*C*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hyper geom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*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)) - (6*C*a^3*cos(c + d*x)^(13/2)*si n(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d* x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(15/2)*sin(c + d*x)*hypergeom([1/2, 1 5/4], 19/4, cos(c + d*x)^2))/(15*d*(sin(c + d*x)^2)^(1/2))
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=a^{3} \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) b \right ) \] Input:
int(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
a**3*(int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a + int(sqrt(cos(c + d*x))*co s(c + d*x)**6,x)*c + int(sqrt(cos(c + d*x))*cos(c + d*x)**5,x)*b + 3*int(s qrt(cos(c + d*x))*cos(c + d*x)**5,x)*c + int(sqrt(cos(c + d*x))*cos(c + d* x)**4,x)*a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*b + 3*int(sqrt(co s(c + d*x))*cos(c + d*x)**4,x)*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)** 3,x)*a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a + int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*b)