Integrand size = 35, antiderivative size = 622 \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (40 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d}+\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
2/11*a*A*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/99*a*(14* A*b+11*B*a)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(9/2)+2/693*(81 *A*a^2+113*A*b^2+209*B*a*b)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d/cos(d*x+c) ^(7/2)+2/3465*(1145*A*a^2*b+15*A*b^3+539*B*a^3+825*B*a*b^2)*sin(d*x+c)*(a+ b*cos(d*x+c))^(1/2)/a/d/cos(d*x+c)^(5/2)+2/3465*(675*A*a^4+1025*A*a^2*b^2- 20*A*b^4+1793*B*a^3*b+55*B*a*b^3)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a^2/d/ cos(d*x+c)^(3/2)+2/3465*(a-b)*(3705*A*a^4*b+255*A*a^2*b^3+40*A*b^5+1617*B* a^5+3069*B*a^3*b^2-110*B*a*b^4)*cot(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2 )/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-sec (d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d+2/3465*(a-b)*(4 0*A*b^4+3*a^4*(225*A-539*B)-6*a^3*b*(505*A-209*B)+15*a^2*b^2*(19*A-121*B)+ 10*a*b^3*(3*A-11*B))*cot(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/ 2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-sec(d*x+c))/(a +b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/d
Result contains complex when optimal does not.
Time = 7.13 (sec) , antiderivative size = 1640, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx =\text {Too large to display} \]
((-4*a*(675*a^6*A - 390*a^4*A*b^2 - 245*a^2*A*b^4 - 40*A*b^6 + 1254*a^5*b* B - 1364*a^3*b^3*B + 110*a*b^5*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[ c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(- 3705*a^5*A*b - 255*a^3*A*b^3 - 40*a*A*b^5 - 1617*a^6*B - 3069*a^4*b^2*B + 110*a^2*b^4*B)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*C sc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)] *Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi [-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]] , (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Co s[c + d*x]])) + 2*(-3705*a^4*A*b^2 - 255*a^2*A*b^4 - 40*A*b^6 - 1617*a^5*b *B - 3069*a^3*b^3*B + 110*a*b^5*B)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(- a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a...
Time = 3.39 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3468, 27, 3042, 3526, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2}{11} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b (6 a A+11 b B) \cos ^2(c+d x)+\left (9 A a^2+22 b B a+11 A b^2\right ) \cos (c+d x)+a (14 A b+11 a B)\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b (6 a A+11 b B) \cos ^2(c+d x)+\left (9 A a^2+22 b B a+11 A b^2\right ) \cos (c+d x)+a (14 A b+11 a B)\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (6 a A+11 b B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (9 A a^2+22 b B a+11 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (14 A b+11 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {3 b \left (22 B a^2+46 A b a+33 b^2 B\right ) \cos ^2(c+d x)+\left (77 B a^3+233 A b a^2+297 b^2 B a+99 A b^3\right ) \cos (c+d x)+a \left (81 A a^2+209 b B a+113 A b^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {3 b \left (22 B a^2+46 A b a+33 b^2 B\right ) \cos ^2(c+d x)+\left (77 B a^3+233 A b a^2+297 b^2 B a+99 A b^3\right ) \cos (c+d x)+a \left (81 A a^2+209 b B a+113 A b^2\right )}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {3 b \left (22 B a^2+46 A b a+33 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^3+233 A b a^2+297 b^2 B a+99 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (81 A a^2+209 b B a+113 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \int \frac {4 a b \left (81 A a^2+209 b B a+113 A b^2\right ) \cos ^2(c+d x)+a \left (405 A a^3+1507 b B a^2+1531 A b^2 a+693 b^3 B\right ) \cos (c+d x)+a \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right )}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {4 a b \left (81 A a^2+209 b B a+113 A b^2\right ) \cos ^2(c+d x)+a \left (405 A a^3+1507 b B a^2+1531 A b^2 a+693 b^3 B\right ) \cos (c+d x)+a \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right )}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {4 a b \left (81 A a^2+209 b B a+113 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (405 A a^3+1507 b B a^2+1531 A b^2 a+693 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {2 \int \frac {\left (1617 B a^3+5055 A b a^2+6655 b^2 B a+2305 A b^3\right ) \cos (c+d x) a^2+2 b \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right ) \cos ^2(c+d x) a+3 \left (675 A a^4+1793 b B a^3+1025 A b^2 a^2+55 b^3 B a-20 A b^4\right ) a}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (1617 B a^3+5055 A b a^2+6655 b^2 B a+2305 A b^3\right ) \cos (c+d x) a^2+2 b \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right ) \cos ^2(c+d x) a+3 \left (675 A a^4+1793 b B a^3+1025 A b^2 a^2+55 b^3 B a-20 A b^4\right ) a}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (1617 B a^3+5055 A b a^2+6655 b^2 B a+2305 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+2 b \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (675 A a^4+1793 b B a^3+1025 A b^2 a^2+55 b^3 B a-20 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {2 \int \frac {3 \left (\left (675 A a^4+2871 b B a^3+3315 A b^2 a^2+1705 b^3 B a+10 A b^4\right ) \cos (c+d x) a^2+\left (1617 B a^5+3705 A b a^4+3069 b^2 B a^3+255 A b^3 a^2-110 b^4 B a+40 A b^5\right ) a\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\left (675 A a^4+2871 b B a^3+3315 A b^2 a^2+1705 b^3 B a+10 A b^4\right ) \cos (c+d x) a^2+\left (1617 B a^5+3705 A b a^4+3069 b^2 B a^3+255 A b^3 a^2-110 b^4 B a+40 A b^5\right ) a}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\left (675 A a^4+2871 b B a^3+3315 A b^2 a^2+1705 b^3 B a+10 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (1617 B a^5+3705 A b a^4+3069 b^2 B a^3+255 A b^3 a^2-110 b^4 B a+40 A b^5\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {a (a-b) \left (3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+a \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {a (a-b) \left (3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {a \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}}{a}}{5 a}}{7 a}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
(2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos [c + d*x]^(9/2)) + ((2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*(1145*a^2*A*b + 15*A *b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5* d*Cos[c + d*x]^(5/2)) + (((2*(a - b)*Sqrt[a + b]*(3705*a^4*A*b + 255*a^2*A *b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Cot[c + d*x]* EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]]) ], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + S ec[c + d*x]))/(a - b)])/(a*d) + (2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 3*a^4*( 225*A - 539*B) - 6*a^3*b*(505*A - 209*B) + 15*a^2*b^2*(19*A - 121*B) + 10* a*b^3*(3*A - 11*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]] /(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d)/a + (2*(675*a^ 4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Co s[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)))/(5*a))/(7*a))/9)/11
3.5.18.3.1 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[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
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_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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] && NeQ[A, B]
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_.) + (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)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(7346\) vs. \(2(572)=1144\).
Time = 37.18 (sec) , antiderivative size = 7347, normalized size of antiderivative = 11.81
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(7347\) |
| default | \(\text {Expression too large to display}\) | \(7451\) |
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
integral((B*b^2*cos(d*x + c)^3 + A*a^2 + (2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(1 3/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{13/2}} \,d x \]