Integrand size = 41, antiderivative size = 462 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=-\frac {b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 a^5 d}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
-b*(12*A*b^6-12*B*a^5*b+15*B*a^3*b^3-6*B*a*b^5-a^2*b^4*(29*A-2*C)+5*a^4*b^ 2*(4*A-C)+6*a^6*C)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/ (a-b)^(5/2)/(a+b)^(5/2)/d+1/2*(12*A*b^2-6*B*a*b+a^2*(A+2*C))*arctanh(sin(d *x+c))/a^5/d-1/2*(12*A*b^5-2*a^5*B+11*a^3*b^2*B-6*a*b^4*B+a^4*b*(6*A-5*C)- a^2*b^3*(21*A-2*C))*tan(d*x+c)/a^4/(a^2-b^2)^2/d+1/2*(6*A*b^4+6*B*a^3*b-3* B*a*b^3+a^4*(A-4*C)-a^2*b^2*(10*A-C))*sec(d*x+c)*tan(d*x+c)/a^3/(a^2-b^2)^ 2/d+1/2*(A*b^2-a*(B*b-C*a))*sec(d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d *x+c))^2+1/2*(7*A*a^2*b^2-4*A*b^4-5*B*a^3*b+2*B*a*b^3+3*C*a^4)*sec(d*x+c)* tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
Time = 5.53 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.31 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {16 b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B+5 a^4 b^2 (4 A-C)+6 a^6 C+a^2 b^4 (-29 A+2 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}-8 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (4 a^7 A-30 a^5 A b^2+68 a^3 A b^4-36 a A b^6+8 a^6 b B-32 a^4 b^3 B+18 a^2 b^5 B+12 a^5 b^2 C-6 a^3 b^4 C+\left (-16 a^6 A b-36 A b^7+8 a^7 B-10 a^5 b^2 B-25 a^3 b^4 B+18 a b^6 B+a^2 b^5 (47 A-6 C)+a^4 b^3 (14 A+15 C)\right ) \cos (c+d x)+2 a b \left (-18 A b^5+4 a^5 B-16 a^3 b^2 B+9 a b^4 B+a^2 b^3 (32 A-3 C)+a^4 (-11 A b+6 b C)\right ) \cos (2 (c+d x))-6 a^4 A b^3 \cos (3 (c+d x))+21 a^2 A b^5 \cos (3 (c+d x))-12 A b^7 \cos (3 (c+d x))+2 a^5 b^2 B \cos (3 (c+d x))-11 a^3 b^4 B \cos (3 (c+d x))+6 a b^6 B \cos (3 (c+d x))+5 a^4 b^3 C \cos (3 (c+d x))-2 a^2 b^5 C \cos (3 (c+d x))\right ) \sec (c+d x) \tan (c+d x)}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{16 a^5 d} \]
((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2]) /Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 8*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(12*A*b^2 - 6*a*b*B + a ^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(4*a^7*A - 3 0*a^5*A*b^2 + 68*a^3*A*b^4 - 36*a*A*b^6 + 8*a^6*b*B - 32*a^4*b^3*B + 18*a^ 2*b^5*B + 12*a^5*b^2*C - 6*a^3*b^4*C + (-16*a^6*A*b - 36*A*b^7 + 8*a^7*B - 10*a^5*b^2*B - 25*a^3*b^4*B + 18*a*b^6*B + a^2*b^5*(47*A - 6*C) + a^4*b^3 *(14*A + 15*C))*Cos[c + d*x] + 2*a*b*(-18*A*b^5 + 4*a^5*B - 16*a^3*b^2*B + 9*a*b^4*B + a^2*b^3*(32*A - 3*C) + a^4*(-11*A*b + 6*b*C))*Cos[2*(c + d*x) ] - 6*a^4*A*b^3*Cos[3*(c + d*x)] + 21*a^2*A*b^5*Cos[3*(c + d*x)] - 12*A*b^ 7*Cos[3*(c + d*x)] + 2*a^5*b^2*B*Cos[3*(c + d*x)] - 11*a^3*b^4*B*Cos[3*(c + d*x)] + 6*a*b^6*B*Cos[3*(c + d*x)] + 5*a^4*b^3*C*Cos[3*(c + d*x)] - 2*a^ 2*b^5*C*Cos[3*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(a + b *Cos[c + d*x])^2))/(16*a^5*d)
Time = 3.20 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.06, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 3534, 25, 3042, 3534, 25, 3042, 3534, 27, 3042, 3534, 25, 3042, 3480, 3042, 3138, 218, 4257}
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.
\(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {\int -\frac {\left (-3 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)+2 a (A b+C b-a B) \cos (c+d x)+2 \left (-\left ((A-C) a^2\right )-b B a+2 A b^2\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {\left (-3 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)+2 a (A b+C b-a B) \cos (c+d x)+2 \left (-\left ((A-C) a^2\right )-b B a+2 A b^2\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-3 \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a (A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (-\left ((A-C) a^2\right )-b B a+2 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {\left (2 \left (3 C a^4-5 b B a^3+7 A b^2 a^2+2 b^3 B a-4 A b^4\right ) \cos ^2(c+d x)+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x)+2 \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {\left (2 \left (3 C a^4-5 b B a^3+7 A b^2 a^2+2 b^3 B a-4 A b^4\right ) \cos ^2(c+d x)+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x)+2 \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {2 \left (3 C a^4-5 b B a^3+7 A b^2 a^2+2 b^3 B a-4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\int -\frac {2 \left (-2 B a^5+b (6 A-5 C) a^4+11 b^2 B a^3-b^3 (21 A-2 C) a^2-6 b^4 B a+\left (-\left ((A+2 C) a^4\right )+4 b B a^3-b^2 (4 A+C) a^2-b^3 B a+2 A b^4\right ) \cos (c+d x) a+12 A b^5-b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{2 a}+\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\int \frac {\left (-2 B a^5+b (6 A-5 C) a^4+11 b^2 B a^3-b^3 (21 A-2 C) a^2-6 b^4 B a+\left (-\left ((A+2 C) a^4\right )+4 b B a^3-b^2 (4 A+C) a^2-b^3 B a+2 A b^4\right ) \cos (c+d x) a+12 A b^5-b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\int \frac {-2 B a^5+b (6 A-5 C) a^4+11 b^2 B a^3-b^3 (21 A-2 C) a^2-6 b^4 B a+\left (-\left ((A+2 C) a^4\right )+4 b B a^3-b^2 (4 A+C) a^2-b^3 B a+2 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+12 A b^5-b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\int -\frac {\left (\left ((A+2 C) a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\int \frac {\left (\left ((A+2 C) a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\int \frac {\left ((A+2 C) a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3480 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right ) \int \sec (c+d x)dx}{a}-\frac {b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\tan (c+d x) \sec (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\tan (c+d x) \left (-2 a^5 B+a^4 (6 A b-5 b C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right ) \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (-(((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x] ))) - (((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(a*d) - (-(((-2*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6 *a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b] *Sqrt[a + b]*d) + ((a^2 - b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*ArcT anh[Sin[c + d*x]])/(a*d))/a) + ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b ^4*B - a^2*b^3*(21*A - 2*C) + a^4*(6*A*b - 5*b*C))*Tan[c + d*x])/(a*d))/a) /(a*(a^2 - b^2)))/(2*a*(a^2 - b^2))
3.10.100.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(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]
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[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.48 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-A \,a^{2}-12 A \,b^{2}+6 B a b -2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5}}-\frac {2 b \left (\frac {-\frac {\left (10 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b -B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (10 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b +B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{2}}+\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}-12 B \,a^{5} b +15 B \,a^{3} b^{3}-6 B a \,b^{5}+6 a^{6} C -5 a^{4} b^{2} C +2 a^{2} C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (A \,a^{2}+12 A \,b^{2}-6 B a b +2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5}}}{d}\) | \(550\) |
default | \(\frac {\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-A \,a^{2}-12 A \,b^{2}+6 B a b -2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5}}-\frac {2 b \left (\frac {-\frac {\left (10 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b -B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (10 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b +B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{2}}+\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}-12 B \,a^{5} b +15 B \,a^{3} b^{3}-6 B a \,b^{5}+6 a^{6} C -5 a^{4} b^{2} C +2 a^{2} C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (A \,a^{2}+12 A \,b^{2}-6 B a b +2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5}}}{d}\) | \(550\) |
risch | \(\text {Expression too large to display}\) | \(2997\) |
1/d*(1/2*A/a^3/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(-A*a-6*A*b+2*B*a)/a^4/(tan(1/ 2*d*x+1/2*c)-1)+1/2/a^5*(-A*a^2-12*A*b^2+6*B*a*b-2*C*a^2)*ln(tan(1/2*d*x+1 /2*c)-1)-2*b/a^5*((-1/2*(10*A*a^2*b^2+A*a*b^3-6*A*b^4-8*B*a^3*b-B*a^2*b^2+ 4*B*a*b^3+6*C*a^4+C*a^3*b-2*C*a^2*b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d *x+1/2*c)^3-1/2*b*a*(10*A*a^2*b^2-A*a*b^3-6*A*b^4-8*B*a^3*b+B*a^2*b^2+4*B* a*b^3+6*C*a^4-C*a^3*b-2*C*a^2*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c))/(tan( 1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^2+1/2*(20*A*a^4*b^2-29*A*a^ 2*b^4+12*A*b^6-12*B*a^5*b+15*B*a^3*b^3-6*B*a*b^5+6*C*a^6-5*C*a^4*b^2+2*C*a ^2*b^4)/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1 /2*c)/((a-b)*(a+b))^(1/2)))-1/2*A/a^3/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-A*a-6 *A*b+2*B*a)/a^4/(tan(1/2*d*x+1/2*c)+1)+1/2*(A*a^2+12*A*b^2-6*B*a*b+2*C*a^2 )/a^5*ln(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 1395 vs. \(2 (442) = 884\).
Time = 110.98 (sec) , antiderivative size = 2858, normalized size of antiderivative = 6.19 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3, x, algorithm="fricas")
[-1/4*(((6*C*a^6*b^3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A*b^9)*cos(d*x + c)^4 + 2*(6*C*a^7* b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C)*a^3 *b^6 - 6*B*a^2*b^7 + 12*A*a*b^8)*cos(d*x + c)^3 + (6*C*a^8*b - 12*B*a^7*b^ 2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^ 6 + 12*A*a^2*b^7)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)* sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2 )) - (((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6*b^4 + 18*B*a^5* b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a *b^9 - 12*A*b^10)*cos(d*x + c)^4 + 2*((A + 2*C)*a^9*b - 6*B*a^8*b^2 + 3*(3 *A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*C)*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*cos(d*x + c)^3 + ((A + 2 *C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2* C)*a^6*b^4 - 18*B*a^5*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2* b^8)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + (((A + 2*C)*a^8*b^2 - 6*B*a^7 *b^3 + 3*(3*A - 2*C)*a^6*b^4 + 18*B*a^5*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18* B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*cos(d*x + c)^4 + 2*((A + 2*C)*a^9*b - 6*B*a^8*b^2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*C)*a^3*b^7 + 6*B*a^2...
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3, x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1744 vs. \(2 (442) = 884\).
Time = 0.43 (sec) , antiderivative size = 1744, normalized size of antiderivative = 3.77 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3, x, algorithm="giac")
1/2*(2*(6*C*a^6*b - 12*B*a^5*b^2 + 20*A*a^4*b^3 - 5*C*a^4*b^3 + 15*B*a^3*b ^4 - 29*A*a^2*b^5 + 2*C*a^2*b^5 - 6*B*a*b^6 + 12*A*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan( 1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^9 - 2*a^7*b^2 + a^5*b^4)*sqrt(a^2 - b^2)) + 2*(A*a^7*tan(1/2*d*x + 1/2*c)^7 - 2*B*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 4*B*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 1 3*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 2*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 2*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 6*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/2*c)^7 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^5 - 2*B*a^7 *tan(1/2*d*x + 1/2*c)^5 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 4*B*a^6*b*tan (1/2*d*x + 1/2*c)^5 + 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 10*B*a^5*b^2*ta n(1/2*d*x + 1/2*c)^5 - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 26*A*a^4*b^3*t an(1/2*d*x + 1/2*c)^5 + 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 15*C*a^4*b^3 *tan(1/2*d*x + 1/2*c)^5 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 35*B*a^3*b ^4*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 67*A*a...
Time = 19.17 (sec) , antiderivative size = 15951, normalized size of antiderivative = 34.53 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^3*(3*A*a^7 + 36*A*b^7 + 2*B*a^7 - 67*A*a^2*b^5 - 29*A *a^3*b^4 + 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2*b^5 + 35*B*a^3*b^4 + 16*B* a^4*b^3 - 10*B*a^5*b^2 + 6*C*a^2*b^5 + 3*C*a^3*b^4 - 15*C*a^4*b^3 - 6*C*a^ 5*b^2 + 18*A*a*b^6 - 4*A*a^6*b - 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) + (tan(c/2 + (d*x)/2)^5*(3*A*a^7 - 36*A*b^7 - 2*B*a^ 7 + 67*A*a^2*b^5 - 29*A*a^3*b^4 - 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2*b^5 - 35*B*a^3*b^4 + 16*B*a^4*b^3 + 10*B*a^5*b^2 - 6*C*a^2*b^5 + 3*C*a^3*b^4 + 15*C*a^4*b^3 - 6*C*a^5*b^2 + 18*A*a*b^6 + 4*A*a^6*b + 18*B*a*b^6 - 4*B*a ^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 + (d*x)/2)^7*(A*a^ 6 - 12*A*b^6 - 2*B*a^6 + 23*A*a^2*b^4 - 10*A*a^3*b^3 - 8*A*a^4*b^2 - 3*B*a ^2*b^4 - 12*B*a^3*b^3 + 4*B*a^4*b^2 - 2*C*a^2*b^4 + C*a^3*b^3 + 6*C*a^4*b^ 2 + 6*A*a*b^5 + 5*A*a^5*b + 6*B*a*b^5 + 2*B*a^5*b))/((a^4*b - a^5)*(a + b) ^2) + (tan(c/2 + (d*x)/2)*(A*a^6 - 12*A*b^6 + 2*B*a^6 + 23*A*a^2*b^4 + 10* A*a^3*b^3 - 8*A*a^4*b^2 + 3*B*a^2*b^4 - 12*B*a^3*b^3 - 4*B*a^4*b^2 - 2*C*a ^2*b^4 - C*a^3*b^3 + 6*C*a^4*b^2 - 6*A*a*b^5 - 5*A*a^5*b + 6*B*a*b^5 + 2*B *a^5*b))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)))/(d*(2*a*b - tan(c/2 + (d*x)/ 2)^4*(2*a^2 - 6*b^2) - tan(c/2 + (d*x)/2)^2*(4*a*b + 4*b^2) + tan(c/2 + (d *x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (atan(((((((4*(4*A*a^21 + 8*C*a^21 - 48*A*a^10*b^11 + 24*A*a^11* b^10 + 212*A*a^12*b^9 - 100*A*a^13*b^8 - 360*A*a^14*b^7 + 164*A*a^15*b^...