Integrand size = 33, antiderivative size = 522 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {\left (20 A b^9-a^2 b^7 (69 A-2 C)-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-8 a^8 b C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {\left (20 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 a^6 d}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:
(20*A*b^9-a^2*b^7*(69*A-2*C)-8*a^6*b^3*(5*A-C)+7*a^4*b^5*(12*A-C)-8*a^8*b* C)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/(a-b)^(1/2)/(a+b )^(1/2)/(a^2-b^2)^3/d+1/2*(20*A*b^2+a^2*(A+2*C))*arctanh(sin(d*x+c))/a^6/d +1/6*b*(60*A*b^6-a^6*(24*A-26*C)+a^4*b^2*(146*A-17*C)-a^2*b^4*(167*A-6*C)) *tan(d*x+c)/a^5/(a^2-b^2)^3/d-1/2*(10*A*b^6-a^6*(A-6*C)+a^4*b^2*(23*A-2*C) -a^2*b^4*(27*A-C))*sec(d*x+c)*tan(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2+C*a^ 2)*sec(d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^3-1/6*(5*A*b^4-4*a ^4*C-a^2*b^2*(10*A+C))*sec(d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d* x+c))^2+1/6*(20*A*b^6-a^2*b^4*(53*A-2*C)+12*a^6*C+a^4*b^2*(48*A+C))*sec(d* x+c)*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
Time = 8.59 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.42 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {\left (C+A \sec ^2(c+d x)\right ) \left (\frac {96 b \left (20 A b^8+7 a^4 b^4 (12 A-C)-8 a^8 C+8 a^6 b^2 (-5 A+C)+a^2 b^6 (-69 A+2 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right ) \cos ^2(c+d x)}{\left (-a^2+b^2\right )^{7/2}}-48 \left (20 A b^2+a^2 (A+2 C)\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 \left (20 A b^2+a^2 (A+2 C)\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (24 a^{10} A-324 a^8 A b^2+1116 a^6 A b^4-830 a^4 A b^6-61 a^2 A b^8+180 A b^{10}+144 a^8 b^2 C-50 a^6 b^4 C-7 a^4 b^6 C+18 a^2 b^8 C-6 a b \left (20 a^8 A-150 A b^8+3 a^6 b^2 (3 A-20 C)+5 a^2 b^6 (80 A-3 C)+3 a^4 b^4 (-103 A+15 C)\right ) \cos (c+d x)+12 b^2 \left (20 A b^8-3 a^8 (7 A-4 C)+a^6 b^2 (85 A-2 C)+a^2 b^6 (-19 A+2 C)-a^4 b^4 (55 A+2 C)\right ) \cos (2 (c+d x))-138 a^7 A b^3 \cos (3 (c+d x))+738 a^5 A b^5 \cos (3 (c+d x))-840 a^3 A b^7 \cos (3 (c+d x))+300 a A b^9 \cos (3 (c+d x))+120 a^7 b^3 C \cos (3 (c+d x))-90 a^5 b^5 C \cos (3 (c+d x))+30 a^3 b^7 C \cos (3 (c+d x))-24 a^6 A b^4 \cos (4 (c+d x))+146 a^4 A b^6 \cos (4 (c+d x))-167 a^2 A b^8 \cos (4 (c+d x))+60 A b^{10} \cos (4 (c+d x))+26 a^6 b^4 C \cos (4 (c+d x))-17 a^4 b^6 C \cos (4 (c+d x))+6 a^2 b^8 C \cos (4 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}\right )}{48 a^6 d (2 A+C+C \cos (2 (c+d x)))} \] Input:
Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x ]
Output:
((C + A*Sec[c + d*x]^2)*((96*b*(20*A*b^8 + 7*a^4*b^4*(12*A - C) - 8*a^8*C + 8*a^6*b^2*(-5*A + C) + a^2*b^6*(-69*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2)/(-a^2 + b^2)^(7/2) - 48*(20*A*b ^2 + a^2*(A + 2*C))*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2] ] + 48*(20*A*b^2 + a^2*(A + 2*C))*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Si n[(c + d*x)/2]] + (2*a*(24*a^10*A - 324*a^8*A*b^2 + 1116*a^6*A*b^4 - 830*a ^4*A*b^6 - 61*a^2*A*b^8 + 180*A*b^10 + 144*a^8*b^2*C - 50*a^6*b^4*C - 7*a^ 4*b^6*C + 18*a^2*b^8*C - 6*a*b*(20*a^8*A - 150*A*b^8 + 3*a^6*b^2*(3*A - 20 *C) + 5*a^2*b^6*(80*A - 3*C) + 3*a^4*b^4*(-103*A + 15*C))*Cos[c + d*x] + 1 2*b^2*(20*A*b^8 - 3*a^8*(7*A - 4*C) + a^6*b^2*(85*A - 2*C) + a^2*b^6*(-19* A + 2*C) - a^4*b^4*(55*A + 2*C))*Cos[2*(c + d*x)] - 138*a^7*A*b^3*Cos[3*(c + d*x)] + 738*a^5*A*b^5*Cos[3*(c + d*x)] - 840*a^3*A*b^7*Cos[3*(c + d*x)] + 300*a*A*b^9*Cos[3*(c + d*x)] + 120*a^7*b^3*C*Cos[3*(c + d*x)] - 90*a^5* b^5*C*Cos[3*(c + d*x)] + 30*a^3*b^7*C*Cos[3*(c + d*x)] - 24*a^6*A*b^4*Cos[ 4*(c + d*x)] + 146*a^4*A*b^6*Cos[4*(c + d*x)] - 167*a^2*A*b^8*Cos[4*(c + d *x)] + 60*A*b^10*Cos[4*(c + d*x)] + 26*a^6*b^4*C*Cos[4*(c + d*x)] - 17*a^4 *b^6*C*Cos[4*(c + d*x)] + 6*a^2*b^8*C*Cos[4*(c + d*x)])*Sin[c + d*x])/((a^ 2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(48*a^6*d*(2*A + C + C*Cos[2*(c + d*x )]))
Time = 4.00 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.07, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3535, 25, 3042, 3534, 25, 3042, 3534, 25, 3042, 3534, 27, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {\left (-\left ((3 A-2 C) a^2\right )+3 b (A+C) \cos (c+d x) a+5 A b^2-4 \left (C a^2+A b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\int \frac {\left (-\left ((3 A-2 C) a^2\right )+3 b (A+C) \cos (c+d x) a+5 A b^2-4 \left (C a^2+A b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\int \frac {-\left ((3 A-2 C) a^2\right )+3 b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+5 A b^2-4 \left (C a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\int -\frac {\left (-3 \left (-4 C a^4-b^2 (10 A+C) a^2+5 A b^4\right ) \cos ^2(c+d x)+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \cos (c+d x)+2 \left (3 (A-2 C) a^4-b^2 (18 A-C) a^2+10 A b^4\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {\left (-3 \left (-4 C a^4-b^2 (10 A+C) a^2+5 A b^4\right ) \cos ^2(c+d x)+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \cos (c+d x)+2 \left (3 (A-2 C) a^4-b^2 (18 A-C) a^2+10 A b^4\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-3 \left (-4 C a^4-b^2 (10 A+C) a^2+5 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 (A-2 C) a^4-b^2 (18 A-C) a^2+10 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {\left (-2 \left (12 C a^6+b^2 (48 A+C) a^4-b^4 (53 A-2 C) a^2+20 A b^6\right ) \cos ^2(c+d x)+a b \left (2 (9 A+5 C) a^4-b^2 (8 A-5 C) a^2+5 A b^4\right ) \cos (c+d x)+6 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {\left (-2 \left (12 C a^6+b^2 (48 A+C) a^4-b^4 (53 A-2 C) a^2+20 A b^6\right ) \cos ^2(c+d x)+a b \left (2 (9 A+5 C) a^4-b^2 (8 A-5 C) a^2+5 A b^4\right ) \cos (c+d x)+6 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {-2 \left (12 C a^6+b^2 (48 A+C) a^4-b^4 (53 A-2 C) a^2+20 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a b \left (2 (9 A+5 C) a^4-b^2 (8 A-5 C) a^2+5 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int -\frac {2 \left (-3 b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \cos ^2(c+d x)+a \left (3 (A+2 C) a^6+b^2 (27 A+8 C) a^4-b^4 (25 A-C) a^2+10 A b^6\right ) \cos (c+d x)+b \left (-\left ((24 A-26 C) a^6\right )+b^2 (146 A-17 C) a^4-b^4 (167 A-6 C) a^2+60 A b^6\right )\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{2 a}+\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {\left (60 A b^7-a^2 (167 A-6 C) b^5+a^4 (146 A-17 C) b^3-3 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \cos ^2(c+d x) b-a^6 (24 A b-26 b C)+a \left (3 (A+2 C) a^6+b^2 (27 A+8 C) a^4-b^4 (25 A-C) a^2+10 A b^6\right ) \cos (c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {60 A b^7-a^2 (167 A-6 C) b^5+a^4 (146 A-17 C) b^3-3 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b-a^6 (24 A b-26 b C)+a \left (3 (A+2 C) a^6+b^2 (27 A+8 C) a^4-b^4 (25 A-C) a^2+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\int \frac {3 \left (\left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {3 \int \frac {\left (\left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right ) \int \sec (c+d x)dx}{a}+\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 \left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 \left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \tan (c+d x)}{a d}+\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right ) \text {arctanh}(\sin (c+d x))}{a d}+\frac {2 \left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
Input:
Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]
Output:
((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[ c + d*x])^3) - (((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan [c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (((20*A*b^6 - a^2* b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Sec[c + d*x]*Tan[c + d*x ])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((3*(10*A*b^6 - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Sec[c + d*x]*Tan[c + d*x])/(a *d) - ((3*((2*(20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a ^4*b^5*(12*A - C) - 8*a^8*b*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)^3*(20*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(a*d)))/a + (b*(60*A*b^6 - a^6*(24*A - 26* C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Tan[c + d*x])/(a*d))/ a)/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[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[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 = 3.63 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(\frac {\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-A \,a^{2}-20 A \,b^{2}-2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}+\frac {A \left (a +8 b \right )}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}+18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}+8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\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^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (A \,a^{2}+20 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}+\frac {A \left (a +8 b \right )}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(673\) |
| default | \(\frac {\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-A \,a^{2}-20 A \,b^{2}-2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}+\frac {A \left (a +8 b \right )}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}+18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}+8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\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^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (A \,a^{2}+20 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}+\frac {A \left (a +8 b \right )}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(673\) |
| risch | \(\text {Expression too large to display}\) | \(3033\) |
Input:
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x,method=_RETURNVER BOSE)
Output:
1/d*(1/2*A/a^4/(tan(1/2*d*x+1/2*c)-1)^2+1/2/a^6*(-A*a^2-20*A*b^2-2*C*a^2)* ln(tan(1/2*d*x+1/2*c)-1)+1/2*A*(a+8*b)/a^5/(tan(1/2*d*x+1/2*c)-1)-2/a^6*b* ((-1/2*(30*A*a^4*b^2+6*A*a^3*b^3-34*A*a^2*b^4-3*A*a*b^5+12*A*b^6+12*C*a^6+ 4*C*a^5*b-6*C*a^4*b^2-C*a^3*b^3+2*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^ 2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(45*A*a^4*b^2-53*A*a^2*b^4+18*A*b^6+18*C*a ^6-11*C*a^4*b^2+3*C*a^2*b^4)*a*b/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d *x+1/2*c)^3-1/2*(30*A*a^4*b^2-6*A*a^3*b^3-34*A*a^2*b^4+3*A*a*b^5+12*A*b^6+ 12*C*a^6-4*C*a^5*b-6*C*a^4*b^2+C*a^3*b^3+2*C*a^2*b^4)*a*b/(a+b)/(a^3-3*a^2 *b+3*a*b^2-b^3)*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)^3+1/2*(40*A*a^6*b^2-84*A*a^4*b^4+69*A*a^2*b^6-20*A*b^8+8*C*a ^8-8*C*a^6*b^2+7*C*a^4*b^4-2*C*a^2*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((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^4/(tan(1/2*d*x+1/2*c)+1)^2+1/2*(A*a^2+20*A*b^2+2*C*a^2)/a^6*ln(tan(1/2 *d*x+1/2*c)+1)+1/2*A*(a+8*b)/a^5/(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 1600 vs. \(2 (499) = 998\).
Time = 47.59 (sec) , antiderivative size = 3269, normalized size of antiderivative = 6.26 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm= "fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm= "maxima")
Output:
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. 1070 vs. \(2 (499) = 998\).
Time = 0.25 (sec) , antiderivative size = 1070, normalized size of antiderivative = 2.05 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm= "giac")
Output:
1/6*(6*(8*C*a^8*b + 40*A*a^6*b^3 - 8*C*a^6*b^3 - 84*A*a^4*b^5 + 7*C*a^4*b^ 5 + 69*A*a^2*b^7 - 2*C*a^2*b^7 - 20*A*b^9)*(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^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt(a^2 - b^2)) + 2*(36*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*C*a^7*b^3*tan(1/2*d *x + 1/2*c)^5 + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^6*b^4*tan(1/2* d*x + 1/2*c)^5 - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 45*C*a^5*b^5*tan(1 /2*d*x + 1/2*c)^5 - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*tan( 1/2*d*x + 1/2*c)^5 + 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^3*b^7*t an(1/2*d*x + 1/2*c)^5 - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*b^8* tan(1/2*d*x + 1/2*c)^5 - 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5 + 36*A*b^10*tan (1/2*d*x + 1/2*c)^5 + 72*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 + 180*A*a^6*b^4* tan(1/2*d*x + 1/2*c)^3 - 116*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 - 392*A*a^4* b^6*tan(1/2*d*x + 1/2*c)^3 + 56*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 + 284*A*a ^2*b^8*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 - 72*A *b^10*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^8*b^2*tan(1/2*d*x + 1/2*c) + 60*C*a^ 7*b^3*tan(1/2*d*x + 1/2*c) + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c) - 6*C*a^6*b ^4*tan(1/2*d*x + 1/2*c) + 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c) - 45*C*a^5*b^ 5*tan(1/2*d*x + 1/2*c) - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) - 6*C*a^4*b^6*t an(1/2*d*x + 1/2*c) - 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^3*b^7...
Time = 10.56 (sec) , antiderivative size = 14213, normalized size of antiderivative = 27.23 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b*cos(c + d*x))^4),x)
Output:
((tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^8 - 59*A*a^2*b^6 - 27*A*a^3*b^5 + 57* A*a^4*b^4 + 21*A*a^5*b^3 - 11*A*a^6*b^2 + 2*C*a^2*b^6 + C*a^3*b^5 - 6*C*a^ 4*b^4 - 4*C*a^5*b^3 + 12*C*a^6*b^2 + 10*A*a*b^7 - 7*A*a^7*b))/(a^5*(a + b) *(a - b)^3) + (tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 59*A*a^2*b^6 + 27* A*a^3*b^5 + 57*A*a^4*b^4 - 21*A*a^5*b^3 - 11*A*a^6*b^2 + 2*C*a^2*b^6 - C*a ^3*b^5 - 6*C*a^4*b^4 + 4*C*a^5*b^3 + 12*C*a^6*b^2 - 10*A*a*b^7 + 7*A*a^7*b ))/(a^5*(a + b)^3*(a - b)) - (2*tan(c/2 + (d*x)/2)^3*(120*A*b^9 - 6*A*a^9 - 364*A*a^2*b^7 - 71*A*a^3*b^6 + 369*A*a^4*b^5 + 45*A*a^5*b^4 - 111*A*a^6* b^3 - 3*A*a^7*b^2 + 12*C*a^2*b^7 + 3*C*a^3*b^6 - 37*C*a^4*b^5 - 8*C*a^5*b^ 4 + 60*C*a^6*b^3 + 30*A*a*b^8 + 21*A*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) + (2*tan(c/2 + (d*x)/2)^7*(6*A*a^9 + 120*A*b^9 - 364*A*a^2*b^7 + 71*A*a^3*b ^6 + 369*A*a^4*b^5 - 45*A*a^5*b^4 - 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*C*a^2 *b^7 - 3*C*a^3*b^6 - 37*C*a^4*b^5 + 8*C*a^5*b^4 + 60*C*a^6*b^3 - 30*A*a*b^ 8 + 21*A*a^8*b))/(3*a^5*(a + b)^3*(a - b)^2) + (2*tan(c/2 + (d*x)/2)^5*(9* A*a^10 + 180*A*b^10 - 611*A*a^2*b^8 + 740*A*a^4*b^6 - 324*A*a^6*b^4 + 36*A *a^8*b^2 + 18*C*a^2*b^8 - 62*C*a^4*b^6 + 110*C*a^6*b^4 - 36*C*a^8*b^2))/(3 *a^5*(a + b)^3*(a - b)^3))/(d*(tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^2*b - 2 *a^3 + 10*b^3) - tan(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a^3 + 5*b^3) + tan(c/2 + (d*x)/2)^6*(6*a*b^2 + 6*a^2*b - 2*a^3 - 10*b^3) + 3*a*b^2 + 3*a^ 2*b + a^3 + b^3 + tan(c/2 + (d*x)/2)^10*(3*a*b^2 - 3*a^2*b + a^3 - b^3)...
Time = 0.24 (sec) , antiderivative size = 9075, normalized size of antiderivative = 17.39 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x)
Output:
( - 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq rt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*a**7*b**4*c - 240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))* cos(c + d*x)*sin(c + d*x)**4*a**6*b**6 + 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*a**5*b**6*c + 504*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - ta n((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*a**4*b** 8 - 42*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq rt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*a**3*b**8*c - 414*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))* cos(c + d*x)*sin(c + d*x)**4*a**2*b**10 + 12*sqrt(a**2 - b**2)*atan((tan(( c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*a*b**10*c + 120*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan ((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*b**12 + 1 44*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a **2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**9*b**2*c + 720*sqrt(a**2 - b* *2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos( c + d*x)*sin(c + d*x)**2*a**8*b**4 - 48*sqrt(a**2 - b**2)*atan((tan((c + d *x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x )**2*a**7*b**4*c - 1032*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - ta...