Integrand size = 43, antiderivative size = 294 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \left (3 a^3 B+15 a b^2 B+5 b^3 (A-C)+3 a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (21 a^2 b B+21 b^3 B+21 a b^2 (A+3 C)+a^3 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (24 A b^2+63 a b B+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (24 A b^3+21 a^3 B+98 a b^2 B+21 a^2 b (3 A+5 C)\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {2 (6 A b+7 a B) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:
-2/5*(3*B*a^3+15*B*a*b^2+5*b^3*(A-C)+3*a^2*b*(3*A+5*C))*EllipticE(sin(1/2* d*x+1/2*c),2^(1/2))/d+2/21*(21*B*a^2*b+21*B*b^3+21*a*b^2*(A+3*C)+a^3*(5*A+ 7*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/105*a*(24*A*b^2+63*B*a*b+ 5*a^2*(5*A+7*C))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/35*(24*A*b^3+21*B*a^3+98* B*a*b^2+21*a^2*b*(3*A+5*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/35*(6*A*b+7*B* a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/7*A*(a+b*cos(d*x+c)) ^3*sin(d*x+c)/d/cos(d*x+c)^(7/2)
Time = 9.52 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (-21 \left (3 a^3 B+15 a b^2 B+5 b^3 (A-C)+3 a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (21 a^2 b B+21 b^3 B+21 a b^2 (A+3 C)+a^3 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {15 a^3 A \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}+\frac {21 a^2 (3 A b+a B) \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {5 a \left (21 A b^2+21 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {21 \left (5 A b^3+3 a^3 B+15 a b^2 B+3 a^2 b (3 A+5 C)\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{105 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(9/2),x]
Output:
(2*(-21*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Ellip ticE[(c + d*x)/2, 2] + 5*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3 *(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (15*a^3*A*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*a^2*(3*A*b + a*B)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*a *(21*A*b^2 + 21*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 3*a^2*b*(3*A + 5*C))*Sin[c + d*x]) /Sqrt[Cos[c + d*x]]))/(105*d)
Time = 1.83 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{7} \int \frac {(a+b \cos (c+d x))^2 \left (-b (A-7 C) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+6 A b+7 a B\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(a+b \cos (c+d x))^2 \left (-b (A-7 C) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+6 A b+7 a B\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b (A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+7 b B+7 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 A b+7 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x)) \left (5 (5 A+7 C) a^2+63 b B a+24 A b^2-b (11 A b-35 C b+7 a B) \cos ^2(c+d x)+\left (21 B a^2+38 A b a+70 b C a+35 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x)) \left (5 (5 A+7 C) a^2+63 b B a+24 A b^2-b (11 A b-35 C b+7 a B) \cos ^2(c+d x)+\left (21 B a^2+38 A b a+70 b C a+35 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 (5 A+7 C) a^2+63 b B a+24 A b^2-b (11 A b-35 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 B a^2+38 A b a+70 b C a+35 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int -\frac {-3 b^2 (11 A b-35 C b+7 a B) \cos ^2(c+d x)+5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right ) \cos (c+d x)+3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {-3 b^2 (11 A b-35 C b+7 a B) \cos ^2(c+d x)+5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right ) \cos (c+d x)+3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {-3 b^2 (11 A b-35 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (2 \int \frac {5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right )-21 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right )-21 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right )-21 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right )}{d}+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right )}{d}+\frac {6 \sin (c+d x) \left (21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*( 6*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2 )) + ((2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(3*d*Co s[c + d*x]^(3/2)) + ((-42*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b* (3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(21*a^2*b*B + 21*b^3*B + 2 1*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d + (6*(24 *A*b^3 + 21*a^3*B + 98*a*b^2*B + 21*a^2*b*(3*A + 5*C))*Sin[c + d*x])/(d*Sq rt[Cos[c + d*x]]))/3)/5)/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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. \(1177\) vs. \(2(277)=554\).
Time = 6.96 (sec) , antiderivative size = 1178, normalized size of antiderivative = 4.01
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1178\) |
| parts | \(\text {Expression too large to display}\) | \(1217\) |
Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x, method=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*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*(-1/56*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*si n(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*(3*A*b^2+3*B*a*b+C*a^2)*(-1/6*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+1 /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)))+2/5*a^2*(3*A*b+B*a)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c )^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*cos(1/2*d*x+1/2*c)* sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2* c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-24* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2 )*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1 /2*d*x+1/2*c)^2+8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-3*(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...
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{3} + 21 i \, B a^{2} b + 21 i \, {\left (A + 3 \, C\right )} a b^{2} + 21 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{3} - 21 i \, B a^{2} b - 21 i \, {\left (A + 3 \, C\right )} a b^{2} - 21 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{3} + 3 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b + 15 i \, B a b^{2} + 5 i \, {\left (A - C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{3} - 3 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b - 15 i \, B a b^{2} - 5 i \, {\left (A - C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (15 \, A a^{3} + 21 \, {\left (3 \, B a^{3} + 3 \, {\left (3 \, A + 5 \, C\right )} a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{3} + 21 \, B a^{2} b + 21 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9 /2),x, algorithm="fricas")
Output:
-1/105*(5*sqrt(2)*(I*(5*A + 7*C)*a^3 + 21*I*B*a^2*b + 21*I*(A + 3*C)*a*b^2 + 21*I*B*b^3)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c)) + 5*sqrt(2)*(-I*(5*A + 7*C)*a^3 - 21*I*B*a^2*b - 21*I*(A + 3 *C)*a*b^2 - 21*I*B*b^3)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(3*I*B*a^3 + 3*I*(3*A + 5*C)*a^2*b + 1 5*I*B*a*b^2 + 5*I*(A - C)*b^3)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-3*I*B *a^3 - 3*I*(3*A + 5*C)*a^2*b - 15*I*B*a*b^2 - 5*I*(A - C)*b^3)*cos(d*x + c )^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c))) - 2*(15*A*a^3 + 21*(3*B*a^3 + 3*(3*A + 5*C)*a^2*b + 15*B*a*b^2 + 5*A*b^3)*cos(d*x + c)^3 + 5*((5*A + 7*C)*a^3 + 21*B*a^2*b + 21*A*a*b^2) *cos(d*x + c)^2 + 21*(B*a^3 + 3*A*a^2*b)*cos(d*x + c))*sqrt(cos(d*x + c))* sin(d*x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)* *(9/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9 /2),x, algorithm="maxima")
Output:
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)^(9/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9 /2),x, algorithm="giac")
Output:
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)^(9/2), x)
Time = 3.75 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.50 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2),x)
Output:
(2*(C*b^3*ellipticE(c/2 + (d*x)/2, 2) + 3*C*a*b^2*ellipticF(c/2 + (d*x)/2, 2)))/d + ((2*A*a^3*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x) ^2))/7 + 2*A*b^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, c os(c + d*x)^2) + 2*A*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*A*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*B*b^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*B*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)) + (2*C*a^3*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (6 *B*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*B*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)) + (6*C*a^2*b*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))
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \,b^{2} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) b^{4}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{4}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{3} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{3} c +6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{2} b^{2}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{2} b c +4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a \,b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) b^{3} c \] Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x)
Output:
3*int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a*b**2*c + int(sqrt(cos(c + d*x)) /cos(c + d*x),x)*b**4 + int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a**4 + 4 *int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**3*b + int(sqrt(cos(c + d*x)) /cos(c + d*x)**3,x)*a**3*c + 6*int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a **2*b**2 + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a**2*b*c + 4*int(sq rt(cos(c + d*x))/cos(c + d*x)**2,x)*a*b**3 + int(sqrt(cos(c + d*x)),x)*b** 3*c