Integrand size = 45, antiderivative size = 334 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)} \]
2/143*a*(5*A+13*B)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2 /13*A*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d/cos(d*x+c)^(13/2)+2/9009*a^3*(22 24*A+2522*B+2717*C)*sin(d*x+c)/d/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(1/2)+2 /15015*a^3*(8368*A+9230*B+10439*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos( d*x+c))^(1/2)+8/45045*a^3*(8368*A+9230*B+10439*C)*sin(d*x+c)/d/cos(d*x+c)^ (3/2)/(a+a*cos(d*x+c))^(1/2)+16/45045*a^3*(8368*A+9230*B+10439*C)*sin(d*x+ c)/d/cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2)+2/1287*a^2*(136*A+182*B+143*C )*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(9/2)
Time = 1.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.67 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (343612 A+325910 B+322751 C+70 (5552 A+5083 B+4576 C) \cos (c+d x)+14 (30334 A+31850 B+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+138450 B \cos (3 (c+d x))+141570 C \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+138450 B \cos (4 (c+d x))+156585 C \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+18460 B \cos (5 (c+d x))+20878 C \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+18460 B \cos (6 (c+d x))+20878 C \cos (6 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{180180 d \cos ^{\frac {13}{2}}(c+d x)} \]
Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x] ^2))/Cos[c + d*x]^(15/2),x]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 325910*B + 322751*C + 70*(5552 *A + 5083*B + 4576*C)*Cos[c + d*x] + 14*(30334*A + 31850*B + 32747*C)*Cos[ 2*(c + d*x)] + 125520*A*Cos[3*(c + d*x)] + 138450*B*Cos[3*(c + d*x)] + 141 570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 138450*B*Cos[4*(c + d *x)] + 156585*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 18460*B*Cos[ 5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 18460 *B*Cos[6*(c + d*x)] + 20878*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(180180* d*Cos[c + d*x]^(13/2))
Time = 1.97 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {3042, 3522, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3459, 3042, 3251, 3042, 3251, 3042, 3250}
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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{15/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^{5/2} (a (5 A+13 B)+a (6 A+13 C) \cos (c+d x))}{2 \cos ^{\frac {13}{2}}(c+d x)}dx}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^{5/2} (a (5 A+13 B)+a (6 A+13 C) \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)}dx}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (5 A+13 B)+a (6 A+13 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {(\cos (c+d x) a+a)^{3/2} \left ((136 A+182 B+143 C) a^2+(96 A+78 B+143 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \frac {(\cos (c+d x) a+a)^{3/2} \left ((136 A+182 B+143 C) a^2+(96 A+78 B+143 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((136 A+182 B+143 C) a^2+(96 A+78 B+143 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((2224 A+2522 B+2717 C) a^3+3 (560 A+598 B+715 C) \cos (c+d x) a^3\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((2224 A+2522 B+2717 C) a^3+3 (560 A+598 B+715 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((2224 A+2522 B+2717 C) a^3+3 (560 A+598 B+715 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \left (\frac {4}{5} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \left (\frac {4}{5} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3250 |
| \(\displaystyle \frac {\frac {2 a^2 (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{11} \left (\frac {2 a^3 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (\frac {2 a^4 (2224 A+2522 B+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {3}{7} a^3 (8368 A+9230 B+10439 C) \left (\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4}{5} \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )\right )\right )}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/C os[c + d*x]^(15/2),x]
(2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2)) + ((2*a^2*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2*a^3*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]] *Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*a^4*(2224*A + 2522*B + 2717* C)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (3*a^ 3*(8368*A + 9230*B + 10439*C)*((2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)* Sqrt[a + a*Cos[c + d*x]]) + (4*((2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2) *Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sq rt[a + a*Cos[c + d*x]])))/5))/7)/9)/11)/(13*a)
3.6.2.3.1 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[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq rt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -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/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 13.31 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {2 a^{2} \sin \left (d x +c \right ) \left (66944 A \left (\cos ^{6}\left (d x +c \right )\right )+73840 B \left (\cos ^{6}\left (d x +c \right )\right )+83512 C \left (\cos ^{6}\left (d x +c \right )\right )+33472 A \left (\cos ^{5}\left (d x +c \right )\right )+36920 B \left (\cos ^{5}\left (d x +c \right )\right )+41756 C \left (\cos ^{5}\left (d x +c \right )\right )+25104 A \left (\cos ^{4}\left (d x +c \right )\right )+27690 B \left (\cos ^{4}\left (d x +c \right )\right )+31317 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+18305 A \left (\cos ^{2}\left (d x +c \right )\right )+14560 B \left (\cos ^{2}\left (d x +c \right )\right )+5005 C \left (\cos ^{2}\left (d x +c \right )\right )+11970 A \cos \left (d x +c \right )+4095 B \cos \left (d x +c \right )+3465 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{45045 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {13}{2}}}\) | \(232\) |
| parts | \(\frac {2 A \sin \left (d x +c \right ) \left (66944 \left (\cos ^{6}\left (d x +c \right )\right )+33472 \left (\cos ^{5}\left (d x +c \right )\right )+25104 \left (\cos ^{4}\left (d x +c \right )\right )+20920 \left (\cos ^{3}\left (d x +c \right )\right )+18305 \left (\cos ^{2}\left (d x +c \right )\right )+11970 \cos \left (d x +c \right )+3465\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{45045 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {13}{2}}}+\frac {2 B \sin \left (d x +c \right ) \left (1136 \left (\cos ^{5}\left (d x +c \right )\right )+568 \left (\cos ^{4}\left (d x +c \right )\right )+426 \left (\cos ^{3}\left (d x +c \right )\right )+355 \left (\cos ^{2}\left (d x +c \right )\right )+224 \cos \left (d x +c \right )+63\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{693 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {11}{2}}}+\frac {2 C \sin \left (d x +c \right ) \left (584 \left (\cos ^{4}\left (d x +c \right )\right )+292 \left (\cos ^{3}\left (d x +c \right )\right )+219 \left (\cos ^{2}\left (d x +c \right )\right )+130 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{315 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {9}{2}}}\) | \(287\) |
int((a+cos(d*x+c)*a)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(15/ 2),x,method=_RETURNVERBOSE)
2/45045*a^2/d*sin(d*x+c)*(66944*A*cos(d*x+c)^6+73840*B*cos(d*x+c)^6+83512* C*cos(d*x+c)^6+33472*A*cos(d*x+c)^5+36920*B*cos(d*x+c)^5+41756*C*cos(d*x+c )^5+25104*A*cos(d*x+c)^4+27690*B*cos(d*x+c)^4+31317*C*cos(d*x+c)^4+20920*A *cos(d*x+c)^3+23075*B*cos(d*x+c)^3+18590*C*cos(d*x+c)^3+18305*A*cos(d*x+c) ^2+14560*B*cos(d*x+c)^2+5005*C*cos(d*x+c)^2+11970*A*cos(d*x+c)+4095*B*cos( d*x+c)+3465*A)*(a*(1+cos(d*x+c)))^(1/2)/(1+cos(d*x+c))/cos(d*x+c)^(13/2)
Time = 0.30 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.57 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 \, {\left (8 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (4184 \, A + 4615 \, B + 3718 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (523 \, A + 416 \, B + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \, {\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(15/2),x, algorithm="fricas")
2/45045*(8*(8368*A + 9230*B + 10439*C)*a^2*cos(d*x + c)^6 + 4*(8368*A + 92 30*B + 10439*C)*a^2*cos(d*x + c)^5 + 3*(8368*A + 9230*B + 10439*C)*a^2*cos (d*x + c)^4 + 5*(4184*A + 4615*B + 3718*C)*a^2*cos(d*x + c)^3 + 35*(523*A + 416*B + 143*C)*a^2*cos(d*x + c)^2 + 315*(38*A + 13*B)*a^2*cos(d*x + c) + 3465*A*a^2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin(d*x + c)/(d*c os(d*x + c)^8 + d*cos(d*x + c)^7)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1004 vs. \(2 (292) = 584\).
Time = 0.41 (sec) , antiderivative size = 1004, normalized size of antiderivative = 3.01 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Too large to display} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(15/2),x, algorithm="maxima")
8/45045*(143*(315*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 945*sq rt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1449*sqrt(2)*a^(5/2)*s in(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1287*sqrt(2)*a^(5/2)*sin(d*x + c)^7/( cos(d*x + c) + 1)^7 + 572*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1 )^9 - 104*sqrt(2)*a^(5/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)*C*(sin(d* x + c)^2/(cos(d*x + c) + 1)^2 + 1)^3/((sin(d*x + c)/(cos(d*x + c) + 1) + 1 )^(11/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(3*sin(d*x + c)^2/( cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + sin(d*x + c) ^6/(cos(d*x + c) + 1)^6 + 1)) + 65*(693*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos( d*x + c) + 1) - 2310*sqrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 4620*sqrt(2)*a^(5/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5478*sqrt(2)*a ^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3575*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 1300*sqrt(2)*a^(5/2)*sin(d*x + c)^11/(cos(d* x + c) + 1)^11 + 200*sqrt(2)*a^(5/2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 )*B*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^4/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(4*sin(d *x + c)^2/(cos(d*x + c) + 1)^2 + 6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4 *sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 1)) + (45045*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 165165*s qrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 414414*sqrt(2)*a^(...
Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(15/2),x, algorithm="giac")
Time = 10.71 (sec) , antiderivative size = 941, normalized size of antiderivative = 2.82 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Too large to display} \]
int(((a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^(15/2),x)
((a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((a^2*(8368 *A + 9230*B + 10439*C)*16i)/(45045*d) - (C*a^2*exp(c*3i + d*x*3i)*8i)/(3*d ) + (C*a^2*exp(c*10i + d*x*10i)*8i)/(3*d) - (a^2*exp(c*5i + d*x*5i)*(6*A + 15*B + 23*C)*16i)/(15*d) + (a^2*exp(c*8i + d*x*8i)*(6*A + 15*B + 23*C)*16 i)/(15*d) + (a^2*exp(c*6i + d*x*6i)*(348*A + 345*B + 379*C)*16i)/(105*d) - (a^2*exp(c*7i + d*x*7i)*(348*A + 345*B + 379*C)*16i)/(105*d) + (a^2*exp(c *4i + d*x*4i)*(1046*A + 1075*B + 1108*C)*16i)/(315*d) - (a^2*exp(c*9i + d* x*9i)*(1046*A + 1075*B + 1108*C)*16i)/(315*d) + (a^2*exp(c*2i + d*x*2i)*(8 368*A + 9230*B + 10439*C)*8i)/(3465*d) - (a^2*exp(c*11i + d*x*11i)*(8368*A + 9230*B + 10439*C)*8i)/(3465*d) - (a^2*exp(c*13i + d*x*13i)*(8368*A + 92 30*B + 10439*C)*16i)/(45045*d)))/((exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x *1i)/2)^(1/2) + exp(c*1i + d*x*1i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d* x*1i)/2)^(1/2) + 6*exp(c*2i + d*x*2i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 6*exp(c*3i + d*x*3i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1 i + d*x*1i)/2)^(1/2) + 15*exp(c*4i + d*x*4i)*(exp(- c*1i - d*x*1i)/2 + exp (c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*5i + d*x*5i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 20*exp(c*6i + d*x*6i)*(exp(- c*1i - d*x*1i) /2 + exp(c*1i + d*x*1i)/2)^(1/2) + 20*exp(c*7i + d*x*7i)*(exp(- c*1i - d*x *1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*8i + d*x*8i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*9i + d*x*9i)*(exp(-...