Integrand size = 42, antiderivative size = 305 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a^2 (13 b B+9 a C) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (7 a^2 B+22 b^2 B+27 a b C\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
-2/15*(7*B*a^3+27*B*a*b^2+27*C*a^2*b+15*C*b^3)*(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+2/21*(15*B*a^ 2*b+7*B*b^3+5*C*a^3+21*C*a*b^2)*(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+2/63*a^2*(13*B*b+9*C*a)*sin( d*x+c)/d/cos(d*x+c)^(7/2)+2/45*a*(7*B*a^2+22*B*b^2+27*C*a*b)*sin(d*x+c)/d/ cos(d*x+c)^(5/2)+2/21*(15*B*a^2*b+7*B*b^3+5*C*a^3+21*C*a*b^2)*sin(d*x+c)/d /cos(d*x+c)^(3/2)+2/9*a*B*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(9/2) +2/15*(7*B*a^3+27*B*a*b^2+27*C*a^2*b+15*C*b^3)*sin(d*x+c)/d/cos(d*x+c)^(1/ 2)
Time = 7.40 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \left (-21 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+15 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {35 a^3 B \sin (c+d x)}{\cos ^{\frac {9}{2}}(c+d x)}+\frac {45 a^2 (3 b B+a C) \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}+\frac {7 a \left (7 a^2 B+27 b^2 B+27 a b C\right ) \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {15 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {21 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{315 d} \]
Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos [c + d*x]^(13/2),x]
(2*(-21*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*EllipticE[(c + d*x) /2, 2] + 15*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*EllipticF[(c + d *x)/2, 2] + (35*a^3*B*Sin[c + d*x])/Cos[c + d*x]^(9/2) + (45*a^2*(3*b*B + a*C)*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (7*a*(7*a^2*B + 27*b^2*B + 27*a*b* C)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (15*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(315*d)
Time = 1.46 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.88, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3508, 3042, 3468, 27, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 3116, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 (B+C \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x)) \left (3 b (a B+3 b C) \cos ^2(c+d x)+\left (7 B a^2+18 b C a+9 b^2 B\right ) \cos (c+d x)+a (13 b B+9 a C)\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x)) \left (3 b (a B+3 b C) \cos ^2(c+d x)+\left (7 B a^2+18 b C a+9 b^2 B\right ) \cos (c+d x)+a (13 b B+9 a C)\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 b (a B+3 b C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 B a^2+18 b C a+9 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (13 b B+9 a C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int -\frac {21 b^2 (a B+3 b C) \cos ^2(c+d x)+9 \left (5 C a^3+15 b B a^2+21 b^2 C a+7 b^3 B\right ) \cos (c+d x)+7 a \left (7 B a^2+27 b C a+22 b^2 B\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 (a B+3 b C) \cos ^2(c+d x)+9 \left (5 C a^3+15 b B a^2+21 b^2 C a+7 b^3 B\right ) \cos (c+d x)+7 a \left (7 B a^2+27 b C a+22 b^2 B\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 (a B+3 b C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (5 C a^3+15 b B a^2+21 b^2 C a+7 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+7 a \left (7 B a^2+27 b C a+22 b^2 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3 \left (15 \left (5 C a^3+15 b B a^2+21 b^2 C a+7 b^3 B\right )+7 \left (7 B a^3+27 b C a^2+27 b^2 B a+15 b^3 C\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \frac {15 \left (5 C a^3+15 b B a^2+21 b^2 C a+7 b^3 B\right )+7 \left (7 B a^3+27 b C a^2+27 b^2 B a+15 b^3 C\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \frac {15 \left (5 C a^3+15 b B a^2+21 b^2 C a+7 b^3 B\right )+7 \left (7 B a^3+27 b C a^2+27 b^2 B a+15 b^3 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)}dx+7 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+7 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {14 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {3}{5} \left (15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
(2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2 *a^2*(13*b*B + 9*a*C)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((14*a*(7*a ^2*B + 22*b^2*B + 27*a*b*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (3*(1 5*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))) + 7*(7*a^3*B + 27* a*b^2*B + 27*a^2*b*C + 15*b^3*C)*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Si n[c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/9
3.9.75.3.1 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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(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)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
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_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1165\) vs. \(2(333)=666\).
Time = 24.98 (sec) , antiderivative size = 1166, normalized size of antiderivative = 3.82
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1166\) |
| parts | \(\text {Expression too large to display}\) | \(1425\) |
int((a+cos(d*x+c)*b)^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x,m ethod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*a^3*(-1/14 4*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)^5-7/180*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)^3-14/15*sin(1/ 2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d* x+1/2*c)^2)^(1/2)+7/15*(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))-7/15*(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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1 /2))))+2*C*b^3/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2 *d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2* d*x+1/2*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)*E llipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+2*b^2*(B*b+3*C*a)*(-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*a^2*(3*B*b+C*a)*(-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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, C a^{3} + 15 i \, B a^{2} b + 21 i \, C a b^{2} + 7 i \, B b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, C a^{3} - 15 i \, B a^{2} b - 21 i \, C a b^{2} - 7 i \, B b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (7 i \, B a^{3} + 27 i \, C a^{2} b + 27 i \, B a b^{2} + 15 i \, C b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-7 i \, B a^{3} - 27 i \, C a^{2} b - 27 i \, B a b^{2} - 15 i \, C b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (21 \, {\left (7 \, B a^{3} + 27 \, C a^{2} b + 27 \, B a b^{2} + 15 \, C b^{3}\right )} \cos \left (d x + c\right )^{4} + 35 \, B a^{3} + 15 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 21 \, C a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (7 \, B a^{3} + 27 \, C a^{2} b + 27 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \]
integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/ 2),x, algorithm="fricas")
-1/315*(15*sqrt(2)*(5*I*C*a^3 + 15*I*B*a^2*b + 21*I*C*a*b^2 + 7*I*B*b^3)*c os(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*C*a^3 - 15*I*B*a^2*b - 21*I*C*a*b^2 - 7*I*B*b^3)*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt (2)*(7*I*B*a^3 + 27*I*C*a^2*b + 27*I*B*a*b^2 + 15*I*C*b^3)*cos(d*x + c)^5* weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-7*I*B*a^3 - 27*I*C*a^2*b - 27*I*B*a*b^2 - 15*I*C*b^ 3)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d* x + c) - I*sin(d*x + c))) - 2*(21*(7*B*a^3 + 27*C*a^2*b + 27*B*a*b^2 + 15* C*b^3)*cos(d*x + c)^4 + 35*B*a^3 + 15*(5*C*a^3 + 15*B*a^2*b + 21*C*a*b^2 + 7*B*b^3)*cos(d*x + c)^3 + 7*(7*B*a^3 + 27*C*a^2*b + 27*B*a*b^2)*cos(d*x + c)^2 + 45*(C*a^3 + 3*B*a^2*b)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/ 2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^3/cos(d *x + c)^(13/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/ 2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^3/cos(d *x + c)^(13/2), x)
Time = 7.26 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {70\,B\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{4},\frac {1}{2};\ -\frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )+210\,B\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+378\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+270\,B\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{315\,d\,{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {\frac {2\,C\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7}+2\,C\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+2\,C\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+\frac {6\,C\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5}}{d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}} \]
(70*B*a^3*sin(c + d*x)*hypergeom([-9/4, 1/2], -5/4, cos(c + d*x)^2) + 210* B*b^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x) ^2) + 378*B*a*b^2*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2) + 270*B*a^2*b*cos(c + d*x)*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(315*d*cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2 )^(1/2)) + ((2*C*a^3*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x )^2))/7 + 2*C*b^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2) + 2*C*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*C*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))