Integrand size = 33, antiderivative size = 523 \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b \left (63 a^4 A b-86 a^2 A b^3+35 A b^5-35 a^5 B+38 a^3 b^2 B-15 a b^4 B\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^4 (a-b)^2 (a+b)^3 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \]
1/4*(24*A*a^4*b-65*A*a^2*b^3+35*A*b^5-8*B*a^5+29*B*a^3*b^2-15*B*a*b^4)*(co s(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))/a^4/(a^2-b^2)^2/d+1/12*(8*A*a^4-61*A*a^2*b^2+35*A*b^4+33*B*a^3*b- 15*B*a*b^3)*(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))/a^3/(a^2-b^2)^2/d+1/4*b*(63*A*a^4*b-86*A*a^2*b^3+3 5*A*b^5-35*B*a^5+38*B*a^3*b^2-15*B*a*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos (1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/a^4/(a-b) ^2/(a+b)^3/d+1/12*(8*A*a^4-61*A*a^2*b^2+35*A*b^4+33*B*a^3*b-15*B*a*b^3)*si n(d*x+c)/a^3/(a^2-b^2)^2/d/cos(d*x+c)^(3/2)+1/2*b*(A*b-B*a)*sin(d*x+c)/a/( a^2-b^2)/d/cos(d*x+c)^(3/2)/(a+b*cos(d*x+c))^2+1/4*b*(13*A*a^2*b-7*A*b^3-9 *B*a^3+3*B*a*b^2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/cos(d*x+c)^(3/2)/(a+b*cos(d *x+c))-1/4*(24*A*a^4*b-65*A*a^2*b^3+35*A*b^5-8*B*a^5+29*B*a^3*b^2-15*B*a*b ^4)*sin(d*x+c)/a^4/(a^2-b^2)^2/d/cos(d*x+c)^(1/2)
Time = 7.13 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.09 \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \left (16 a^6 A+328 a^4 A b^2-641 a^2 A b^4+315 A b^6-168 a^5 b B+285 a^3 b^3 B-135 a b^5 B\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (160 a^5 A b-512 a^3 A b^3+280 a A b^5-48 a^6 B+240 a^4 b^2 B-120 a^2 b^4 B\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{b}+\frac {2 \left (72 a^4 A b^2-195 a^2 A b^4+105 A b^6-24 a^5 b B+87 a^3 b^3 B-45 a b^5 B\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2 \sec (c+d x) (-3 A b \sin (c+d x)+a B \sin (c+d x))}{a^4}+\frac {A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {17 a^2 A b^4 \sin (c+d x)-11 A b^6 \sin (c+d x)-13 a^3 b^3 B \sin (c+d x)+7 a b^5 B \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 A \sec (c+d x) \tan (c+d x)}{3 a^3}\right )}{d} \]
((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 - 168*a^5*b*B + 285*a^3*b^3*B - 135*a*b^5*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((160*a^5*A*b - 512*a^3*A*b^3 + 280*a*A*b^5 - 48*a^6*B + 240*a^4*b ^2*B - 120*a^2*b^4*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b) /(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(72*a^4*A*b^2 - 195*a^2*A*b^4 + 105*A*b^6 - 24*a^5*b*B + 87*a^3*b^3*B - 45*a*b^5*B)*Cos[2*(c + d*x)]*(-2 *a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[Arc Sin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sq rt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]] *((2*Sec[c + d*x]*(-3*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/a^4 + (A*b^4*S in[c + d*x] - a*b^3*B*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x] )^2) + (17*a^2*A*b^4*Sin[c + d*x] - 11*A*b^6*Sin[c + d*x] - 13*a^3*b^3*B*S in[c + d*x] + 7*a*b^5*B*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d
Time = 4.00 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.98, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3479, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3119, 3481, 3042, 3120, 3284}
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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3479 |
| \(\displaystyle \frac {\int \frac {4 A a^2+3 b B a-4 (A b-a B) \cos (c+d x) a-7 A b^2+5 b (A b-a B) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 A a^2+3 b B a-4 (A b-a B) \cos (c+d x) a-7 A b^2+5 b (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 A a^2+3 b B a-4 (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a-7 A b^2+5 b (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a-4 \left (-2 B a^3+4 A b a^2-b^2 B a-A b^3\right ) \cos (c+d x) a+35 A b^4+3 b \left (-9 B a^3+13 A b a^2+3 b^2 B a-7 A b^3\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a-4 \left (-2 B a^3+4 A b a^2-b^2 B a-A b^3\right ) \cos (c+d x) a+35 A b^4+3 b \left (-9 B a^3+13 A b a^2+3 b^2 B a-7 A b^3\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a-4 \left (-2 B a^3+4 A b a^2-b^2 B a-A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^4+3 b \left (-9 B a^3+13 A b a^2+3 b^2 B a-7 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {-b \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)-4 a \left (2 A a^4-12 b B a^3+14 A b^2 a^2+3 b^3 B a-7 A b^4\right ) \cos (c+d x)+3 \left (-8 B a^5+24 A b a^4+29 b^2 B a^3-65 A b^3 a^2-15 b^4 B a+35 A b^5\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-b \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)-4 a \left (2 A a^4-12 b B a^3+14 A b^2 a^2+3 b^3 B a-7 A b^4\right ) \cos (c+d x)+3 \left (-8 B a^5+24 A b a^4+29 b^2 B a^3-65 A b^3 a^2-15 b^4 B a+35 A b^5\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-b \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 a \left (2 A a^4-12 b B a^3+14 A b^2 a^2+3 b^3 B a-7 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^5+24 A b a^4+29 b^2 B a^3-65 A b^3 a^2-15 b^4 B a+35 A b^5\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+4 \left (-6 B a^5+20 A b a^4+30 b^2 B a^3-64 A b^3 a^2-15 b^4 B a+35 A b^5\right ) \cos (c+d x) a+105 A b^6+3 b \left (-8 B a^5+24 A b a^4+29 b^2 B a^3-65 A b^3 a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+4 \left (-6 B a^5+20 A b a^4+30 b^2 B a^3-64 A b^3 a^2-15 b^4 B a+35 A b^5\right ) \cos (c+d x) a+105 A b^6+3 b \left (-8 B a^5+24 A b a^4+29 b^2 B a^3-65 A b^3 a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+4 \left (-6 B a^5+20 A b a^4+30 b^2 B a^3-64 A b^3 a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+105 A b^6+3 b \left (-8 B a^5+24 A b a^4+29 b^2 B a^3-65 A b^3 a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {a \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \cos (c+d x) b^2+\left (8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {a \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \cos (c+d x) b^2+\left (8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {a \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {a \left (8 A a^4+33 b B a^3-61 A b^2 a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (8 A a^6-72 b B a^5+128 A b^2 a^4+99 b^3 B a^3-223 A b^4 a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 b^2 \left (-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}+\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b^2 \left (-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {3 b^2 \left (-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b}+\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\frac {2 \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {6 \left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {\frac {2 a b \left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^2 \left (-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
(b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b* Cos[c + d*x])^2) + ((b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])) + ((2*(8 *a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/ (3*a*d*Cos[c + d*x]^(3/2)) - (-(((6*(24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(c + d*x)/2, 2])/d + ((2* a*b*(8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Elliptic F[(c + d*x)/2, 2])/d + (6*b^2*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a ^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2 ])/((a + b)*d))/b)/a) + (6*(24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]))/(3*a ))/(2*a*(a^2 - b^2)))/(4*a*(a^2 - b^2))
3.4.82.3.1 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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
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[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin [e + f*x])^(1 + n)/(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[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B) *(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n }, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Rat ionalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(I ntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0]) ))
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^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[(-(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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{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]
Leaf count of result is larger than twice the leaf count of optimal. \(2130\) vs. \(2(579)=1158\).
Time = 20.43 (sec) , antiderivative size = 2131, normalized size of antiderivative = 4.07
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a^3*(-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)/(c os(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)*E llipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(-3*A*b+B*a)/a^4/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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1 /2)))+2*(A*b-B*a)*b/a^2*(-1/2/a*b^2/(a^2-b^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)/(2*b*cos(1/2*d*x+1/2*c)^2+a-b) ^2-3/4*b^2*(3*a^2-b^2)/a^2/(a^2-b^2)^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)/(2*b*cos(1/2*d*x+1/2*c)^2+a-b)-7/8/(a +b)/(a^2-b^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)/(-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))+1/4/(a+b)/(a^2-b^2)/a*(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))*b+3/8/(a+b)/(a^2-b^2)/a^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)/(-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))*b^2-9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*...
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\int { \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]