Integrand size = 35, antiderivative size = 286 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a^3 (44 A+35 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (33 A+35 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (143 A+105 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \] Output:
4/5*a^3*(7*A+5*C)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*s ec(d*x+c)^(1/2)/d+4/231*a^3*(143*A+105*C)*cos(d*x+c)^(1/2)*InverseJacobiAM (1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+8/385*a^3*(44*A+35*C)*sin(d*x+c )/d/sec(d*x+c)^(3/2)+2/11*C*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(3/ 2)+4/33*C*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d/sec(d*x+c)^(3/2)+2/231*(33 *A+35*C)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+4/231*a^3*(143 *A+105*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.53 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.80 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (160 (143 A+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-2464 i (7 A+5 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (51744 i A+36960 i C+10 (2354 A+1953 C) \sin (c+d x)+308 (18 A+25 C) \sin (2 (c+d x))+660 A \sin (3 (c+d x))+2835 C \sin (3 (c+d x))+770 C \sin (4 (c+d x))+105 C \sin (5 (c+d x)))\right )}{9240 d} \] Input:
Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x ]],x]
Output:
(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(160*(143*A + 105*C)*Sqrt[ Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(7*A + 5*C)*E^(I*(c + d *x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2 *I)*(c + d*x))] + Cos[c + d*x]*((51744*I)*A + (36960*I)*C + 10*(2354*A + 1 953*C)*Sin[c + d*x] + 308*(18*A + 25*C)*Sin[2*(c + d*x)] + 660*A*Sin[3*(c + d*x)] + 2835*C*Sin[3*(c + d*x)] + 770*C*Sin[4*(c + d*x)] + 105*C*Sin[5*( c + d*x)])))/(9240*d*E^(I*d*x))
Time = 1.89 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.97, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.657, Rules used = {3042, 4709, 3042, 3525, 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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A+C \cos (c+d x)^2\right )}{\sqrt {\sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 \left (C \cos ^2(c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {1}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 (a (11 A+3 C)+6 a C \cos (c+d x))dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 (a (11 A+3 C)+6 a C \cos (c+d x))dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (11 A+3 C)+6 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{9} \int \frac {3}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (11 A+5 C) a^2+(33 A+35 C) \cos (c+d x) a^2\right )dx+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (11 A+5 C) a^2+(33 A+35 C) \cos (c+d x) a^2\right )dx+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (11 A+5 C) a^2+(33 A+35 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {2}{7} \int 3 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (11 A+7 C) a^3+2 (44 A+35 C) \cos (c+d x) a^3\right )dx+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (11 A+7 C) a^3+2 (44 A+35 C) \cos (c+d x) a^3\right )dx+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (11 A+7 C) a^3+2 (44 A+35 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \sqrt {\cos (c+d x)} \left (2 (44 A+35 C) \cos ^2(c+d x) a^4+5 (11 A+7 C) a^4+\left (5 (11 A+7 C) a^4+2 (44 A+35 C) a^4\right ) \cos (c+d x)\right )dx+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (2 (44 A+35 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+5 (11 A+7 C) a^4+\left (5 (11 A+7 C) a^4+2 (44 A+35 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (77 (7 A+5 C) a^4+5 (143 A+105 C) \cos (c+d x) a^4\right )dx+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \int \sqrt {\cos (c+d x)} \left (77 (7 A+5 C) a^4+5 (143 A+105 C) \cos (c+d x) a^4\right )dx+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 (7 A+5 C) a^4+5 (143 A+105 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^4 (7 A+5 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (7 A+5 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 a^4 (143 A+105 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (7 A+5 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 a^4 (143 A+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (7 A+5 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 a^4 (143 A+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 a^4 (7 A+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {2 (33 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}+\frac {6}{7} \left (\frac {4 a^4 (44 A+35 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {154 a^4 (7 A+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+5 a^4 (143 A+105 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {4 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}}{11 a}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\right )\) |
Input:
Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[ c + d*x])^3*Sin[c + d*x])/(11*d) + ((4*C*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos [c + d*x])^2*Sin[c + d*x])/(3*d) + ((2*(33*A + 35*C)*Cos[c + d*x]^(3/2)*(a ^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(7*d) + (6*((4*a^4*(44*A + 35*C)*Cos[ c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + ((154*a^4*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/d + 5*a^4*(143*A + 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)/3)/(11*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_.) + (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))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 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]
Time = 37.47 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (3360 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-14560 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (1320 A +25760 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-4752 A -24080 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (6622 A +13090 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2288 A -2940 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+715 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 )-1617 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 )+525 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 )-1155 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 )}{1155 \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 {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(436\) |
| parts | \(\text {Expression too large to display}\) | \(1186\) |
Input:
int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x,method=_RETUR NVERBOSE)
Output:
-4/1155*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(3360* C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-14560*C*cos(1/2*d*x+1/2*c)*sin( 1/2*d*x+1/2*c)^10+(1320*A+25760*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c) +(-4752*A-24080*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(6622*A+13090*C )*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2288*A-2940*C)*sin(1/2*d*x+1/2 *c)^2*cos(1/2*d*x+1/2*c)+715*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))-1617*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))+525*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))-1155*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 complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (105 \, C a^{3} \cos \left (d x + c\right )^{5} + 385 \, C a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (11 \, A + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 77 \, {\left (9 \, A + 10 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 10 \, {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{1155 \, d} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algori thm="fricas")
Output:
-2/1155*(5*I*sqrt(2)*(143*A + 105*C)*a^3*weierstrassPInverse(-4, 0, cos(d* x + c) + I*sin(d*x + c)) - 5*I*sqrt(2)*(143*A + 105*C)*a^3*weierstrassPInv erse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt(2)*(7*A + 5*C)*a^3 *weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d* x + c))) + 231*I*sqrt(2)*(7*A + 5*C)*a^3*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (105*C*a^3*cos(d*x + c) ^5 + 385*C*a^3*cos(d*x + c)^4 + 15*(11*A + 42*C)*a^3*cos(d*x + c)^3 + 77*( 9*A + 10*C)*a^3*cos(d*x + c)^2 + 10*(143*A + 105*C)*a^3*cos(d*x + c))*sin( d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=a^{3} \left (\int \frac {A}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 A \cos {\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 A \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 C \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 C \cos ^{4}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {C \cos ^{5}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx\right ) \] Input:
integrate((a+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)/sec(d*x+c)**(1/2),x)
Output:
a**3*(Integral(A/sqrt(sec(c + d*x)), x) + Integral(3*A*cos(c + d*x)/sqrt(s ec(c + d*x)), x) + Integral(3*A*cos(c + d*x)**2/sqrt(sec(c + d*x)), x) + I ntegral(A*cos(c + d*x)**3/sqrt(sec(c + d*x)), x) + Integral(C*cos(c + d*x) **2/sqrt(sec(c + d*x)), x) + Integral(3*C*cos(c + d*x)**3/sqrt(sec(c + d*x )), x) + Integral(3*C*cos(c + d*x)**4/sqrt(sec(c + d*x)), x) + Integral(C* cos(c + d*x)**5/sqrt(sec(c + d*x)), x))
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)) , x)
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)) , x)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3)/(1/cos(c + d*x))^(1/2) ,x)
Output:
int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3)/(1/cos(c + d*x))^(1/2) , x)
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\sec \left (d x +c \right )}d x \right ) c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) c \right ) \] Input:
int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x)
Output:
a**3*(int(sqrt(sec(c + d*x))/sec(c + d*x),x)*a + 3*int((sqrt(sec(c + d*x)) *cos(c + d*x))/sec(c + d*x),x)*a + int((sqrt(sec(c + d*x))*cos(c + d*x)**5 )/sec(c + d*x),x)*c + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**4)/sec(c + d *x),x)*c + int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x),x)*a + 3* int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x),x)*c + 3*int((sqrt(s ec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x),x)*a + int((sqrt(sec(c + d*x))* cos(c + d*x)**2)/sec(c + d*x),x)*c)