Integrand size = 35, antiderivative size = 321 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {2 \left (30 a^2 b^2 (A-C)-b^4 (5 A+3 C)+a^4 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {8 a b \left (b^2 (3 A+C)+a^2 (A+3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}-\frac {4 a b \left (2 b^2 (33 A-5 C)+3 a^2 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}-\frac {2 b^2 \left (b^2 (59 A-3 C)+3 a^2 (3 A+5 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 \left (16 A b^2+a^2 (3 A+5 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)} \]
-2/5*(30*a^2*b^2*(A-C)-b^4*(5*A+3*C)+a^4*(3*A+5*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+8/3*a*b* (b^2*(3*A+C)+a^2*(A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)* EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d-2/15*b^2*(b^2*(59*A-3*C)+3*a^2*(3* A+5*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+16/15*A*b*(a+b*cos(d*x+c))^3*sin(d*x +c)/d/cos(d*x+c)^(3/2)+2/5*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(5 /2)+2/5*(16*A*b^2+a^2*(3*A+5*C))*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c )^(1/2)-4/15*a*b*(2*b^2*(33*A-5*C)+3*a^2*(3*A+5*C))*sin(d*x+c)*cos(d*x+c)^ (1/2)/d
Time = 5.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {-6 \left (30 a^2 b^2 (A-C)-b^4 (5 A+3 C)+a^4 (3 A+5 C)\right ) \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 a b \left (b^2 (3 A+C)+a^2 (A+3 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+40 a^3 A b \sin (c+d x)+40 a b^3 C \cos ^2(c+d x) \sin (c+d x)+6 b^4 C \cos ^3(c+d x) \sin (c+d x)+9 a^4 A \sin (2 (c+d x))+90 a^2 A b^2 \sin (2 (c+d x))+15 a^4 C \sin (2 (c+d x))+6 a^4 A \tan (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)} \]
(-6*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Cos[c + d*x]^ (3/2)*EllipticE[(c + d*x)/2, 2] + 40*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*C os[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 40*a^3*A*b*Sin[c + d*x] + 40 *a*b^3*C*Cos[c + d*x]^2*Sin[c + d*x] + 6*b^4*C*Cos[c + d*x]^3*Sin[c + d*x] + 9*a^4*A*Sin[2*(c + d*x)] + 90*a^2*A*b^2*Sin[2*(c + d*x)] + 15*a^4*C*Sin [2*(c + d*x)] + 6*a^4*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))
Time = 2.22 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3527, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3512, 27, 3042, 3502, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {2}{5} \int \frac {(a+b \cos (c+d x))^3 \left (-5 b (A-C) \cos ^2(c+d x)+a (3 A+5 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {(a+b \cos (c+d x))^3 \left (-5 b (A-C) \cos ^2(c+d x)+a (3 A+5 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-5 b (A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (3 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {(a+b \cos (c+d x))^2 \left (-5 b^2 (11 A-3 C) \cos ^2(c+d x)+2 a b (A+15 C) \cos (c+d x)+3 \left ((3 A+5 C) a^2+16 A b^2\right )\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b \cos (c+d x))^2 \left (-5 b^2 (11 A-3 C) \cos ^2(c+d x)+2 a b (A+15 C) \cos (c+d x)+3 \left ((3 A+5 C) a^2+16 A b^2\right )\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-5 b^2 (11 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (A+15 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left ((3 A+5 C) a^2+16 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (2 \int \frac {(a+b \cos (c+d x)) \left (-5 b \left (3 (3 A+5 C) a^2+b^2 (59 A-3 C)\right ) \cos ^2(c+d x)-a \left (3 (3 A+5 C) a^2+b^2 (101 A-45 C)\right ) \cos (c+d x)+2 b \left ((19 A+45 C) a^2+96 A b^2\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {(a+b \cos (c+d x)) \left (-5 b \left (3 (3 A+5 C) a^2+b^2 (59 A-3 C)\right ) \cos ^2(c+d x)-a \left (3 (3 A+5 C) a^2+b^2 (101 A-45 C)\right ) \cos (c+d x)+2 b \left ((19 A+45 C) a^2+96 A b^2\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-5 b \left (3 (3 A+5 C) a^2+b^2 (59 A-3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a \left (3 (3 A+5 C) a^2+b^2 (101 A-45 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left ((19 A+45 C) a^2+96 A b^2\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{5} \int \frac {5 \left (-6 a b \left (3 (3 A+5 C) a^2+2 b^2 (33 A-5 C)\right ) \cos ^2(c+d x)-3 \left ((3 A+5 C) a^4+30 b^2 (A-C) a^2-b^4 (5 A+3 C)\right ) \cos (c+d x)+2 a b \left ((19 A+45 C) a^2+96 A b^2\right )\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {-6 a b \left (3 (3 A+5 C) a^2+2 b^2 (33 A-5 C)\right ) \cos ^2(c+d x)-3 \left ((3 A+5 C) a^4+30 b^2 (A-C) a^2-b^4 (5 A+3 C)\right ) \cos (c+d x)+2 a b \left ((19 A+45 C) a^2+96 A b^2\right )}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {-6 a b \left (3 (3 A+5 C) a^2+2 b^2 (33 A-5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 \left ((3 A+5 C) a^4+30 b^2 (A-C) a^2-b^4 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a b \left ((19 A+45 C) a^2+96 A b^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (20 a b \left ((A+3 C) a^2+b^2 (3 A+C)\right )-3 \left ((3 A+5 C) a^4+30 b^2 (A-C) a^2-b^4 (5 A+3 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {20 a b \left ((A+3 C) a^2+b^2 (3 A+C)\right )-3 \left ((3 A+5 C) a^4+30 b^2 (A-C) a^2-b^4 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {20 a b \left ((A+3 C) a^2+b^2 (3 A+C)\right )-3 \left ((3 A+5 C) a^4+30 b^2 (A-C) a^2-b^4 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (20 a b \left (a^2 (A+3 C)+b^2 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-3 \left (a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (20 a b \left (a^2 (A+3 C)+b^2 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-3 \left (a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (20 a b \left (a^2 (A+3 C)+b^2 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {6 \left (a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {40 a b \left (a^2 (A+3 C)+b^2 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {2 b^2 \left (3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d}-\frac {4 a b \left (3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {6 \left (a^2 (3 A+5 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {6 \left (a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac {5}{2}}(c+d x)}\) |
(2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + ((16* A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ((-6*( 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(c + d*x )/2, 2])/d + (40*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2 , 2])/d - (4*a*b*(2*b^2*(33*A - 5*C) + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x ]]*Sin[c + d*x])/d - (2*b^2*(b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/d + (6*(16*A*b^2 + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))/3)/5
3.7.100.3.1 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[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[(-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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, 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]
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^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 32.46 (sec) , antiderivative size = 1343, normalized size of antiderivative = 4.18
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1343\) |
| default | \(\text {Expression too large to display}\) | \(1595\) |
2*(A*b^4+6*C*a^2*b^2)*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1 /2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Ellipti cE(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+2*(4*A*a*b ^3+4*C*a^3*b)/d*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))-2*(6*A*a^2*b^2+C*a^ 4)*(-2*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)*sin(1/2*d*x+1/2*c)^2+(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(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)*Ellipti cE(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-2/5*C*b^4 *((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1 /2*c)*sin(1/2*d*x+1/2*c)^6+8*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-2*sin (1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-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+1/2*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-2/5*a^4*A*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d *x+1/2*c)^2)^(1/2)/(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)^3*(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*sin(1/2*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {20 \, \sqrt {2} {\left (i \, {\left (A + 3 \, C\right )} a^{3} b + i \, {\left (3 \, A + C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 20 \, \sqrt {2} {\left (-i \, {\left (A + 3 \, C\right )} a^{3} b - i \, {\left (3 \, A + C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {2} {\left (i \, {\left (3 \, A + 5 \, C\right )} a^{4} + 30 i \, {\left (A - C\right )} a^{2} b^{2} - i \, {\left (5 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-i \, {\left (3 \, A + 5 \, C\right )} a^{4} - 30 i \, {\left (A - C\right )} a^{2} b^{2} + i \, {\left (5 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, C b^{4} \cos \left (d x + c\right )^{4} + 20 \, C a b^{3} \cos \left (d x + c\right )^{3} + 20 \, A a^{3} b \cos \left (d x + c\right ) + 3 \, A a^{4} + 3 \, {\left ({\left (3 \, A + 5 \, C\right )} a^{4} + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{3}} \]
-1/15*(20*sqrt(2)*(I*(A + 3*C)*a^3*b + I*(3*A + C)*a*b^3)*cos(d*x + c)^3*w eierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 20*sqrt(2)*(-I* (A + 3*C)*a^3*b - I*(3*A + C)*a*b^3)*cos(d*x + c)^3*weierstrassPInverse(-4 , 0, cos(d*x + c) - I*sin(d*x + c)) + 3*sqrt(2)*(I*(3*A + 5*C)*a^4 + 30*I* (A - C)*a^2*b^2 - I*(5*A + 3*C)*b^4)*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*sqrt(2)*(- I*(3*A + 5*C)*a^4 - 30*I*(A - C)*a^2*b^2 + I*(5*A + 3*C)*b^4)*cos(d*x + c) ^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c))) - 2*(3*C*b^4*cos(d*x + c)^4 + 20*C*a*b^3*cos(d*x + c)^3 + 20*A* a^3*b*cos(d*x + c) + 3*A*a^4 + 3*((3*A + 5*C)*a^4 + 30*A*a^2*b^2)*cos(d*x + c)^2)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Time = 5.16 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2\,A\,b^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,A\,a\,b^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,C\,a^3\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,C\,a\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,C\,a^2\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,A\,a^3\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {12\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*A*b^4*ellipticE(c/2 + (d*x)/2, 2))/d + (8*A*a*b^3*ellipticF(c/2 + (d*x) /2, 2))/d + (8*C*a^3*b*ellipticF(c/2 + (d*x)/2, 2))/d + (4*C*a*b^3*((2*cos (c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (12*C*a^2*b^2*ellipticE(c/2 + (d*x)/2, 2))/d + (2*A*a^4*sin(c + d*x)*hyper geom([-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^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) - (2*C*b^4*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)) + (8*A*a^3*b*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)) + (1 2*A*a^2*b^2*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*c os(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2))