Integrand size = 33, antiderivative size = 326 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {b \left (4 A b^2+a^2 (A+2 C)\right ) x}{a^5}+\frac {2 b^2 \left (5 a^2 A b^2-4 A b^4+3 a^4 C-2 a^2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \] Output:
-b*(4*A*b^2+a^2*(A+2*C))*x/a^5+2*b^2*(5*A*a^2*b^2-4*A*b^4+3*C*a^4-2*C*a^2* b^2)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(3/2)/( a+b)^(3/2)/d-1/3*(12*A*b^4-a^2*b^2*(7*A-6*C)-a^4*(2*A+3*C))*sin(d*x+c)/a^4 /(a^2-b^2)/d+b*(2*A*b^2-a^2*(A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)/d-1 /3*(4*A*b^2-a^2*(A-3*C))*cos(d*x+c)^2*sin(d*x+c)/a^2/(a^2-b^2)/d+(A*b^2+C* a^2)*cos(d*x+c)^2*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))
Time = 1.87 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {-12 b \left (4 A b^2+a^2 (A+2 C)\right ) (c+d x)+\frac {24 b^2 \left (4 A b^4-3 a^4 C+a^2 b^2 (-5 A+2 C)\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+3 a \left (12 A b^2+a^2 (3 A+4 C)\right ) \sin (c+d x)-\frac {12 a b^3 \left (A b^2+a^2 C\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))}-6 a^2 A b \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 a^5 d} \] Input:
Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x ]
Output:
(-12*b*(4*A*b^2 + a^2*(A + 2*C))*(c + d*x) + (24*b^2*(4*A*b^4 - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] )/(a^2 - b^2)^(3/2) + 3*a*(12*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] - (12* a*b^3*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) - 6*a^2*A*b*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^5*d)
Time = 2.36 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4589, 3042, 4592, 3042, 4592, 27, 3042, 4592, 27, 3042, 4407, 3042, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4589 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (-\left ((A-3 C) a^2\right )+b (A+C) \sec (c+d x) a+4 A b^2-3 \left (C a^2+A b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {-\left ((A-3 C) a^2\right )+b (A+C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+4 A b^2-3 \left (C a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\int \frac {\cos ^2(c+d x) \left (-2 b \left (4 A b^2-a^2 (A-3 C)\right ) \sec ^2(c+d x)+a \left ((2 A+3 C) a^2+A b^2\right ) \sec (c+d x)+6 b \left (2 A b^2-a^2 (A-C)\right )\right )}{a+b \sec (c+d x)}dx}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\int \frac {-2 b \left (4 A b^2-a^2 (A-3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left ((2 A+3 C) a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 b \left (2 A b^2-a^2 (A-C)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {2 \cos (c+d x) \left (-\left ((2 A+3 C) a^4\right )-b^2 (7 A-6 C) a^2+b \left ((A+3 C) a^2+2 A b^2\right ) \sec (c+d x) a+12 A b^4-3 b^2 \left (2 A b^2-a^2 (A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {\cos (c+d x) \left (-\left ((2 A+3 C) a^4\right )-b^2 (7 A-6 C) a^2+b \left ((A+3 C) a^2+2 A b^2\right ) \sec (c+d x) a+12 A b^4-3 b^2 \left (2 A b^2-a^2 (A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {-\left ((2 A+3 C) a^4\right )-b^2 (7 A-6 C) a^2+b \left ((A+3 C) a^2+2 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+12 A b^4-3 b^2 \left (2 A b^2-a^2 (A-C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}-\frac {\int -\frac {3 \left (b \left (a^2-b^2\right ) \left ((A+2 C) a^2+4 A b^2\right )-a b^2 \left (2 A b^2-a^2 (A-C)\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \int \frac {b \left (a^2-b^2\right ) \left ((A+2 C) a^2+4 A b^2\right )-a b^2 \left (2 A b^2-a^2 (A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \int \frac {b \left (a^2-b^2\right ) \left ((A+2 C) a^2+4 A b^2\right )-a b^2 \left (2 A b^2-a^2 (A-C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {b^2 \left (-3 a^4 C-a^2 b^2 (5 A-2 C)+4 A b^4\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {b x \left (a^2-b^2\right ) \left (a^2 (A+2 C)+4 A b^2\right )}{a}\right )}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {b^2 \left (-3 a^4 C-a^2 b^2 (5 A-2 C)+4 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {b x \left (a^2-b^2\right ) \left (a^2 (A+2 C)+4 A b^2\right )}{a}\right )}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {b \left (-3 a^4 C-a^2 b^2 (5 A-2 C)+4 A b^4\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}+\frac {b x \left (a^2-b^2\right ) \left (a^2 (A+2 C)+4 A b^2\right )}{a}\right )}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {b \left (-3 a^4 C-a^2 b^2 (5 A-2 C)+4 A b^4\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}+\frac {b x \left (a^2-b^2\right ) \left (a^2 (A+2 C)+4 A b^2\right )}{a}\right )}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {2 b \left (-3 a^4 C-a^2 b^2 (5 A-2 C)+4 A b^4\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}+\frac {b x \left (a^2-b^2\right ) \left (a^2 (A+2 C)+4 A b^2\right )}{a}\right )}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {b x \left (a^2-b^2\right ) \left (a^2 (A+2 C)+4 A b^2\right )}{a}+\frac {2 b^2 \left (-3 a^4 C-a^2 b^2 (5 A-2 C)+4 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}+\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{a d}}{a}}{3 a}}{a \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]
Output:
((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[ c + d*x])) - (((4*A*b^2 - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a *d) - ((3*b*(2*A*b^2 - a^2*(A - C))*Cos[c + d*x]*Sin[c + d*x])/(a*d) - ((3 *((b*(a^2 - b^2)*(4*A*b^2 + a^2*(A + 2*C))*x)/a + (2*b^2*(4*A*b^4 - a^2*b^ 2*(5*A - 2*C) - 3*a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b ]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/a + ((12*A*b^4 - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(a*d))/a)/(3*a))/(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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b ^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/( a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b *Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)* (m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Time = 1.07 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 a A \,b^{2}-a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 a A \,b^{2}-2 a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A +A \,a^{2} b -3 a A \,b^{2}-a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+b \left (a^{2} A +4 A \,b^{2}+2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 b^{2} \left (-\frac {a b \left (A \,b^{2}+C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b^{2}-4 A \,b^{4}+3 a^{4} C -2 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) | \(329\) |
| default | \(\frac {-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 a A \,b^{2}-a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 a A \,b^{2}-2 a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A +A \,a^{2} b -3 a A \,b^{2}-a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+b \left (a^{2} A +4 A \,b^{2}+2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 b^{2} \left (-\frac {a b \left (A \,b^{2}+C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b^{2}-4 A \,b^{4}+3 a^{4} C -2 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) | \(329\) |
| risch | \(-\frac {A b x}{a^{3}}-\frac {4 x \,b^{3} A}{a^{5}}-\frac {2 x b C}{a^{3}}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{2} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,b^{2}}{2 a^{4} d}-\frac {i A b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{3} d}-\frac {3 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{2}}{2 a^{4} d}-\frac {2 i b^{3} \left (A \,b^{2}+C \,a^{2}\right ) \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{5} \left (a^{2}-b^{2}\right ) d \left ({\mathrm e}^{2 i \left (d x +c \right )} a +2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {3 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{2} d}+\frac {i A b \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{3} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A \,b^{4}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {4 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {5 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {4 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {A \sin \left (3 d x +3 c \right )}{12 a^{2} d}\) | \(976\) |
Input:
int(cos(d*x+c)^3*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,x,method=_RETURNVER BOSE)
Output:
1/d*(-2/a^5*(((-A*a^3-A*a^2*b-3*A*a*b^2-C*a^3)*tan(1/2*d*x+1/2*c)^5+(-2/3* a^3*A-6*a*A*b^2-2*a^3*C)*tan(1/2*d*x+1/2*c)^3+(-A*a^3+A*a^2*b-3*A*a*b^2-C* a^3)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^3+b*(A*a^2+4*A*b^2+2*C*a ^2)*arctan(tan(1/2*d*x+1/2*c)))-2*b^2/a^5*(-a*b*(A*b^2+C*a^2)/(a^2-b^2)*ta n(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)-(5*A* a^2*b^2-4*A*b^4+3*C*a^4-2*C*a^2*b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arcta nh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))
Time = 0.16 (sec) , antiderivative size = 989, normalized size of antiderivative = 3.03 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,x, algorithm= "fricas")
Output:
[-1/6*(6*((A + 2*C)*a^7*b + 2*(A - 2*C)*a^5*b^3 - (7*A - 2*C)*a^3*b^5 + 4* A*a*b^7)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^6*b^2 + 2*(A - 2*C)*a^4*b^4 - ( 7*A - 2*C)*a^2*b^6 + 4*A*b^8)*d*x - 3*(3*C*a^4*b^3 + (5*A - 2*C)*a^2*b^5 - 4*A*b^7 + (3*C*a^5*b^2 + (5*A - 2*C)*a^3*b^4 - 4*A*a*b^6)*cos(d*x + c))*s qrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2* sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos( d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*((2*A + 3*C)*a^7*b + (5*A - 9* C)*a^5*b^3 - (19*A - 6*C)*a^3*b^5 + 12*A*a*b^7 + (A*a^8 - 2*A*a^6*b^2 + A* a^4*b^4)*cos(d*x + c)^3 - 2*(A*a^7*b - 2*A*a^5*b^3 + A*a^3*b^5)*cos(d*x + c)^2 + ((2*A + 3*C)*a^8 + 2*(A - 3*C)*a^6*b^2 - (10*A - 3*C)*a^4*b^4 + 6*A *a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 2*a^8*b^2 + a^6*b^4)*d*cos( d*x + c) + (a^9*b - 2*a^7*b^3 + a^5*b^5)*d), -1/3*(3*((A + 2*C)*a^7*b + 2* (A - 2*C)*a^5*b^3 - (7*A - 2*C)*a^3*b^5 + 4*A*a*b^7)*d*x*cos(d*x + c) + 3* ((A + 2*C)*a^6*b^2 + 2*(A - 2*C)*a^4*b^4 - (7*A - 2*C)*a^2*b^6 + 4*A*b^8)* d*x - 3*(3*C*a^4*b^3 + (5*A - 2*C)*a^2*b^5 - 4*A*b^7 + (3*C*a^5*b^2 + (5*A - 2*C)*a^3*b^4 - 4*A*a*b^6)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(- a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - ((2*A + 3*C) *a^7*b + (5*A - 9*C)*a^5*b^3 - (19*A - 6*C)*a^3*b^5 + 12*A*a*b^7 + (A*a^8 - 2*A*a^6*b^2 + A*a^4*b^4)*cos(d*x + c)^3 - 2*(A*a^7*b - 2*A*a^5*b^3 + A*a ^3*b^5)*cos(d*x + c)^2 + ((2*A + 3*C)*a^8 + 2*(A - 3*C)*a^6*b^2 - (10*A...
\[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cos(d*x+c)**3*(A+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**2,x)
Output:
Integral((A + C*sec(c + d*x)**2)*cos(c + d*x)**3/(a + b*sec(c + d*x))**2, x)
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^3*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,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*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.33 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {6 \, {\left (3 \, C a^{4} b^{2} + 5 \, A a^{2} b^{4} - 2 \, C a^{2} b^{4} - 4 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {6 \, {\left (C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {3 \, {\left (A a^{2} b + 2 \, C a^{2} b + 4 \, A b^{3}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}}}{3 \, d} \] Input:
integrate(cos(d*x+c)^3*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,x, algorithm= "giac")
Output:
1/3*(6*(3*C*a^4*b^2 + 5*A*a^2*b^4 - 2*C*a^2*b^4 - 4*A*b^6)*(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*t an(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^7 - a^5*b^2)*sqrt(-a^2 + b^2)) + 6*(C*a^2*b^3*tan(1/2*d*x + 1/2*c) + A*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - a^4*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) - 3*(A*a^2*b + 2*C*a^2*b + 4*A*b^3)*(d*x + c)/a^5 + 2*(3*A*a^2*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 3*A*a*b*tan(1/2*d*x + 1/2*c) ^5 + 9*A*b^2*tan(1/2*d*x + 1/2*c)^5 + 2*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 6*C *a^2*tan(1/2*d*x + 1/2*c)^3 + 18*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*A*a^2*ta n(1/2*d*x + 1/2*c) + 3*C*a^2*tan(1/2*d*x + 1/2*c) - 3*A*a*b*tan(1/2*d*x + 1/2*c) + 9*A*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^4 ))/d
Time = 21.80 (sec) , antiderivative size = 6978, normalized size of antiderivative = 21.40 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^3*(A + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^2,x)
Output:
- ((2*tan(c/2 + (d*x)/2)^3*(A*a^5 + 36*A*b^5 - 3*C*a^5 - 19*A*a^2*b^3 - 7* A*a^3*b^2 + 18*C*a^2*b^3 + 3*C*a^3*b^2 + 6*A*a*b^4 - 8*A*a^4*b - 9*C*a^4*b ))/(3*a^4*(a + b)*(a - b)) - (2*tan(c/2 + (d*x)/2)^5*(A*a^5 - 36*A*b^5 - 3 *C*a^5 + 19*A*a^2*b^3 - 7*A*a^3*b^2 - 18*C*a^2*b^3 + 3*C*a^3*b^2 + 6*A*a*b ^4 + 8*A*a^4*b + 9*C*a^4*b))/(3*a^4*(a + b)*(a - b)) + (2*tan(c/2 + (d*x)/ 2)^7*(A*a^5 + 4*A*b^5 + C*a^5 - 3*A*a^2*b^3 + A*a^3*b^2 + 2*C*a^2*b^3 - C* a^3*b^2 - 2*A*a*b^4 - C*a^4*b))/(a^4*(a + b)*(a - b)) - (2*tan(c/2 + (d*x) /2)*(A*a^5 - 4*A*b^5 + C*a^5 + 3*A*a^2*b^3 + A*a^3*b^2 - 2*C*a^2*b^3 - C*a ^3*b^2 - 2*A*a*b^4 + C*a^4*b))/(a^4*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x)/2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(2*a + 4*b) - tan(c/2 + (d*x)/ 2)^6*(2*a - 4*b) + 6*b*tan(c/2 + (d*x)/2)^4)) - (atan(-((((((32*(2*A*a^11* b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14*b^4 + A*a^15*b ^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^ 2 + A*a^17*b + 2*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (32*ta n(c/2 + (d*x)/2)*(A*b^3*4i + a^2*b*(A + 2*C)*1i)*(2*a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i + a^2*b*(A + 2*C)*1i))/a^5 + (32*tan(c/2 + (d*x)/2)*(32*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10 + 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^7*b^5 + 7*A^2*a^ 8*b^4 - 2*A^2*a^9*b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - ...
Time = 0.17 (sec) , antiderivative size = 1187, normalized size of antiderivative = 3.64 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^3*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,x)
Output:
(18*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)*a**4*b**2*c + 30*sqrt( - a**2 + b**2)*ata n((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*a**3*b**4 - 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)*a**2*b**4*c - 24*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)*a*b**6 + 18*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/ 2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a**3*b**3*c + 30*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a**2*b**5 - 12*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan ((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a*b**5*c - 24*sqrt( - a**2 + b**2)* atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*b**7 - cos(c + d*x)*sin(c + d*x)**3*a**8 + 2*cos(c + d*x)*sin(c + d*x)**3*a**6* b**2 - cos(c + d*x)*sin(c + d*x)**3*a**4*b**4 + 3*cos(c + d*x)*sin(c + d*x )*a**8 + 3*cos(c + d*x)*sin(c + d*x)*a**7*c - 6*cos(c + d*x)*sin(c + d*x)* a**5*b**2*c - 9*cos(c + d*x)*sin(c + d*x)*a**4*b**4 + 3*cos(c + d*x)*sin(c + d*x)*a**3*b**4*c + 6*cos(c + d*x)*sin(c + d*x)*a**2*b**6 - 3*cos(c + d* x)*a**7*b*c - 3*cos(c + d*x)*a**7*b*d*x - 6*cos(c + d*x)*a**6*b*c**2 - 6*c os(c + d*x)*a**6*b*c*d*x - 6*cos(c + d*x)*a**5*b**3*c - 6*cos(c + d*x)*a** 5*b**3*d*x + 12*cos(c + d*x)*a**4*b**3*c**2 + 12*cos(c + d*x)*a**4*b**3...