Integrand size = 43, antiderivative size = 538 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B-a^2 b^3 (65 A-3 C)+a^4 (24 A b-9 b C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 a^5 \left (a^2-b^2\right )^2 d}-\frac {b \left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^5 (a-b)^2 (a+b)^3 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))} \]
-1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B-a^2*b^3*(65*A-3*C)+a^4*(24* A*b-9*C*b))*(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))/a^4/(a^2-b^2)^2/d+1/12*(105*A*b^6-72*a^5*b*B+99*a^ 3*b^3*B-45*a*b^5*B+a^4*b^2*(128*A-15*C)-a^2*b^4*(223*A-9*C)+8*a^6*(A+3*C)) *(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2 *c),2^(1/2))/a^5/(a^2-b^2)^2/d-1/4*b*(35*A*b^6-35*a^5*b*B+38*a^3*b^3*B-15* a*b^5*B-a^2*b^4*(86*A-3*C)+3*a^4*b^2*(21*A-2*C)+15*a^6*C)*(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*a/(a+b),2^ (1/2))/a^5/(a-b)^2/(a+b)^3/d+1/2*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^(5/2)*sin( d*x+c)/a/(a^2-b^2)/d/(b+a*cos(d*x+c))^2-1/4*(7*A*b^4+9*B*a^3*b-3*B*a*b^3-5 *a^4*C-a^2*b^2*(13*A+C))*cos(d*x+c)^(3/2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(b+ a*cos(d*x+c))+1/12*(35*A*b^4+33*B*a^3*b-15*B*a*b^3+a^4*(8*A-21*C)-a^2*b^2* (61*A-3*C))*sin(d*x+c)*cos(d*x+c)^(1/2)/a^3/(a^2-b^2)^2/d
Time = 9.83 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {2 \left (-56 a^4 A b+73 a^2 A b^3-35 A b^5+24 a^5 B-21 a^3 b^2 B+15 a b^4 B-15 a^4 b C-3 a^2 b^3 C\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (16 a^5 A+112 a^3 A b^2-56 a A b^4-96 a^4 b B+24 a^2 b^3 B+48 a^5 C+24 a^3 b^2 C\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{a}+\frac {2 \left (-72 a^4 A b+195 a^2 A b^3-105 A b^5+24 a^5 B-87 a^3 b^2 B+45 a b^4 B+27 a^4 b C-9 a^2 b^3 C\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a^2 b \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{48 a^3 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2 A \sin (c+d x)}{3 a^3}-\frac {-A b^4 \sin (c+d x)+a b^3 B \sin (c+d x)-a^2 b^2 C \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}+\frac {-17 a^2 A b^3 \sin (c+d x)+11 A b^5 \sin (c+d x)+13 a^3 b^2 B \sin (c+d x)-7 a b^4 B \sin (c+d x)-9 a^4 b C \sin (c+d x)+3 a^2 b^3 C \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 (b+a \cos (c+d x))}\right )}{d} \]
Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
((2*(-56*a^4*A*b + 73*a^2*A*b^3 - 35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15* a*b^4*B - 15*a^4*b*C - 3*a^2*b^3*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + ((16*a^5*A + 112*a^3*A*b^2 - 56*a*A*b^4 - 96*a^4*b*B + 24*a ^2*b^3*B + 48*a^5*C + 24*a^3*b^2*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*El lipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(-72*a^4*A*b + 1 95*a^2*A*b^3 - 105*A*b^5 + 24*a^5*B - 87*a^3*b^2*B + 45*a*b^4*B + 27*a^4*b *C - 9*a^2*b^3*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d *x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x]) /(a^2*b*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(48*a^3*(a - b) ^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*((2*A*Sin[c + d*x])/(3*a^3) - (-(A*b ^4*Sin[c + d*x]) + a*b^3*B*Sin[c + d*x] - a^2*b^2*C*Sin[c + d*x])/(2*a^3*( a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (-17*a^2*A*b^3*Sin[c + d*x] + 11*A*b^ 5*Sin[c + d*x] + 13*a^3*b^2*B*Sin[c + d*x] - 7*a*b^4*B*Sin[c + d*x] - 9*a^ 4*b*C*Sin[c + d*x] + 3*a^2*b^3*C*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*(b + a *Cos[c + d*x]))))/d
Time = 3.71 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4600, 3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^{3/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )}{(a+b \sec (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-\left ((4 A-3 C) a^2\right )-3 b B a+7 A b^2\right ) \cos ^2(c+d x)\right )-4 a (A b+C b-a B) \cos (c+d x)+5 \left (A b^2-a (b B-a C)\right )\right )}{2 (b+a \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-\left ((4 A-3 C) a^2\right )-3 b B a+7 A b^2\right ) \cos ^2(c+d x)\right )-4 a (A b+C b-a B) \cos (c+d x)+5 \left (A b^2-a (b B-a C)\right )\right )}{(b+a \cos (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left ((4 A-3 C) a^2+3 b B a-7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 a (A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left (A b^2-a (b B-a C)\right )\right )}{\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {\int -\frac {\sqrt {\cos (c+d x)} \left (-\left (\left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)\right )-4 a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x)+3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right )\right )}{2 (b+a \cos (c+d x))}dx}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\left (\left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)\right )-4 a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x)+3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right )\right )}{b+a \cos (c+d x)}dx}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (-\left ((8 A-21 C) a^4\right )-33 b B a^3+b^2 (61 A-3 C) a^2+15 b^3 B a-35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right )\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {-\frac {\frac {2 \int -\frac {-3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)-4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \cos (c+d x)+b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )}{2 \sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)-4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \cos (c+d x)+b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx}{a}-\frac {\int -\frac {a b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {a b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx}{a}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {a b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {a b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right )+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a}-\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{a}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right )}{a d}-\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d}}{3 a}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac {-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}-\frac {-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d}-\frac {\frac {\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right )}{a d}-\frac {6 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a d (a+b)}}{a}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2) *d*(b + a*Cos[c + d*x])^2) + (-(((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4* C - a^2*b^2*(13*A + C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d* (b + a*Cos[c + d*x]))) - (-1/3*((-6*(35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 1 5*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*EllipticE[(c + d*x )/2, 2])/(a*d) + ((2*(105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*Ellipti cF[(c + d*x)/2, 2])/(a*d) - (6*b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 1 5*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Elli pticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d))/a)/a - (2*(35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[ Cos[c + d*x]]*Sin[c + d*x])/(3*a*d))/(2*a*(a^2 - b^2)))/(4*a*(a^2 - b^2))
3.14.29.3.1 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_)]), 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[(-(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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(2288\) vs. \(2(598)=1196\).
Time = 5.98 (sec) , antiderivative size = 2289, normalized size of antiderivative = 4.25
int(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/a^5*b^2*(5*A *b^2-4*B*a*b+3*C*a^2)*(a^2/b/(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*cos(1/2*d*x+1/2*c)^2*a-a+b)-1/2/(a +b)/b*(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*s in(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))+1/2*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)*El lipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/b/(a^2-b^2 )/(a^2-a*b)*a^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)*EllipticPi(cos(1 /2*d*x+1/2*c),2*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^2-a*b)*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)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/ 2)))-2*b^3*(A*b^2-B*a*b+C*a^2)/a^5*(1/2*a^2/b/(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*cos(1/2*d*x+1/2*c )^2*a-a+b)^2+3/4*a^2*(a^2-3*b^2)/b^2/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*si n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*a-a +b)-3/8/(a+b)/(a^2-b^2)/b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*...
\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^3,x, algorithm="fricas")
integral((C*cos(d*x + c)*sec(d*x + c)^2 + B*cos(d*x + c)*sec(d*x + c) + A* cos(d*x + c))*sqrt(cos(d*x + c))/(b^3*sec(d*x + c)^3 + 3*a*b^2*sec(d*x + c )^2 + 3*a^2*b*sec(d*x + c) + a^3), x)
Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^3,x, algorithm="maxima")
\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^3,x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*cos(d*x + c)^(3/2)/(b*se c(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]