Integrand size = 33, antiderivative size = 237 \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=-\frac {\left (40 a^4-12 a^2 b^2-b^4\right ) x}{8 b^6}+\frac {2 a^3 \left (5 a^2-4 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^6 \sqrt {a+b} d}+\frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}-\frac {\left (20 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))} \]
-1/8*(40*a^4-12*a^2*b^2-b^4)*x/b^6+1/3*a*(15*a^2-2*b^2)*sin(d*x+c)/b^5/d-1 /8*(20*a^2-b^2)*cos(d*x+c)*sin(d*x+c)/b^4/d+5/3*a*cos(d*x+c)^2*sin(d*x+c)/ b^3/d-5/4*cos(d*x+c)^3*sin(d*x+c)/b^2/d+cos(d*x+c)^4*sin(d*x+c)/b/d/(a+b*c os(d*x+c))+2*a^3*(5*a^2-4*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b) ^(1/2))/b^6/d/(a-b)^(1/2)/(a+b)^(1/2)
Time = 4.58 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {-\frac {384 a^3 \left (5 a^2-4 b^2\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {-960 a^5 c+288 a^3 b^2 c+24 a b^4 c-960 a^5 d x+288 a^3 b^2 d x+24 a b^4 d x+24 b \left (-40 a^4+12 a^2 b^2+b^4\right ) (c+d x) \cos (c+d x)+24 a^2 b \left (40 a^2-7 b^2\right ) \sin (c+d x)+240 a^3 b^2 \sin (2 (c+d x))-32 a b^4 \sin (2 (c+d x))-40 a^2 b^3 \sin (3 (c+d x))-3 b^5 \sin (3 (c+d x))+10 a b^4 \sin (4 (c+d x))-3 b^5 \sin (5 (c+d x))}{a+b \cos (c+d x)}}{192 b^6 d} \]
((-384*a^3*(5*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (-960*a^5*c + 288*a^3*b^2*c + 24*a*b^4*c - 960*a ^5*d*x + 288*a^3*b^2*d*x + 24*a*b^4*d*x + 24*b*(-40*a^4 + 12*a^2*b^2 + b^4 )*(c + d*x)*Cos[c + d*x] + 24*a^2*b*(40*a^2 - 7*b^2)*Sin[c + d*x] + 240*a^ 3*b^2*Sin[2*(c + d*x)] - 32*a*b^4*Sin[2*(c + d*x)] - 40*a^2*b^3*Sin[3*(c + d*x)] - 3*b^5*Sin[3*(c + d*x)] + 10*a*b^4*Sin[4*(c + d*x)] - 3*b^5*Sin[5* (c + d*x)])/(a + b*Cos[c + d*x]))/(192*b^6*d)
Time = 1.90 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.38, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3527, 25, 3042, 3529, 25, 3042, 3528, 25, 3042, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (1-\sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int -\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-5 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-5 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (4 \left (a^2-b^2\right )-5 \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^2(c+d x) \left (-20 a \left (a^2-b^2\right ) \cos ^2(c+d x)-b \left (a^2-b^2\right ) \cos (c+d x)+15 a \left (a^2-b^2\right )\right )}{a+b \cos (c+d x)}dx}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\cos ^2(c+d x) \left (-20 a \left (a^2-b^2\right ) \cos ^2(c+d x)-b \left (a^2-b^2\right ) \cos (c+d x)+15 a \left (a^2-b^2\right )\right )}{a+b \cos (c+d x)}dx}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-20 a \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 a \left (a^2-b^2\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {\cos (c+d x) \left (40 \left (a^2-b^2\right ) a^2-5 b \left (a^2-b^2\right ) \cos (c+d x) a-3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\cos (c+d x) \left (40 \left (a^2-b^2\right ) a^2-5 b \left (a^2-b^2\right ) \cos (c+d x) a-3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (40 \left (a^2-b^2\right ) a^2-5 b \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int -\frac {-8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)-b \left (a^2-b^2\right ) \left (20 a^2+3 b^2\right ) \cos (c+d x)+3 a \left (20 a^4-21 b^2 a^2+b^4\right )}{a+b \cos (c+d x)}dx}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {-8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)-b \left (a^2-b^2\right ) \left (20 a^2+3 b^2\right ) \cos (c+d x)+3 a \left (20 a^4-21 b^2 a^2+b^4\right )}{a+b \cos (c+d x)}dx}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {-8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (a^2-b^2\right ) \left (20 a^2+3 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (20 a^4-21 b^2 a^2+b^4\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {\int \frac {3 \left (a b \left (20 a^4-21 b^2 a^2+b^4\right )+\left (40 a^6-52 b^2 a^4+11 b^4 a^2+b^6\right ) \cos (c+d x)\right )}{a+b \cos (c+d x)}dx}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {3 \int \frac {a b \left (20 a^4-21 b^2 a^2+b^4\right )+\left (40 a^6-52 b^2 a^4+11 b^4 a^2+b^6\right ) \cos (c+d x)}{a+b \cos (c+d x)}dx}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {3 \int \frac {a b \left (20 a^4-21 b^2 a^2+b^4\right )+\left (40 a^6-52 b^2 a^4+11 b^4 a^2+b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {3 \left (\frac {x \left (40 a^6-52 a^4 b^2+11 a^2 b^4+b^6\right )}{b}-\frac {8 a^3 \left (5 a^4-9 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}\right )}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {3 \left (\frac {x \left (40 a^6-52 a^4 b^2+11 a^2 b^4+b^6\right )}{b}-\frac {8 a^3 \left (5 a^4-9 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\frac {3 \left (\frac {x \left (40 a^6-52 a^4 b^2+11 a^2 b^4+b^6\right )}{b}-\frac {16 a^3 \left (5 a^4-9 a^2 b^2+4 b^4\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {5 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b d}-\frac {-\frac {20 a \left (a^2-b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d}-\frac {-\frac {3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {3 \left (\frac {x \left (40 a^6-52 a^4 b^2+11 a^2 b^4+b^6\right )}{b}-\frac {16 a^3 \left (5 a^4-9 a^2 b^2+4 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}\right )}{b}-\frac {8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)}{b d}}{2 b}}{3 b}}{4 b}}{b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}\) |
(Cos[c + d*x]^4*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x])) + ((-5*(a^2 - b^2 )*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d) - ((-20*a*(a^2 - b^2)*Cos[c + d*x]^ 2*Sin[c + d*x])/(3*b*d) - ((-3*(a^2 - b^2)*(20*a^2 - b^2)*Cos[c + d*x]*Sin [c + d*x])/(2*b*d) - ((3*(((40*a^6 - 52*a^4*b^2 + 11*a^2*b^4 + b^6)*x)/b - (16*a^3*(5*a^4 - 9*a^2*b^2 + 4*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2]) /Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)))/b - (8*a*(15*a^2 - 2*b^2)*( a^2 - b^2)*Sin[c + d*x])/(b*d))/(2*b))/(3*b))/(4*b))/(b*(a^2 - b^2))
3.7.1.3.1 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_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
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[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*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] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 2.45 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a^{3} \left (\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (5 a^{2}-4 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{6}}-\frac {2 \left (\frac {\left (-4 a^{3} b -\frac {3}{2} a^{2} b^{2}-\frac {1}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12 a^{3} b -\frac {3}{2} a^{2} b^{2}+\frac {8}{3} a \,b^{3}+\frac {7}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12 a^{3} b +\frac {3}{2} a^{2} b^{2}-\frac {7}{8} b^{4}+\frac {8}{3} a \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-4 a^{3} b +\frac {3}{2} a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (40 a^{4}-12 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{6}}}{d}\) | \(301\) |
| default | \(\frac {\frac {2 a^{3} \left (\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (5 a^{2}-4 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{6}}-\frac {2 \left (\frac {\left (-4 a^{3} b -\frac {3}{2} a^{2} b^{2}-\frac {1}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12 a^{3} b -\frac {3}{2} a^{2} b^{2}+\frac {8}{3} a \,b^{3}+\frac {7}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12 a^{3} b +\frac {3}{2} a^{2} b^{2}-\frac {7}{8} b^{4}+\frac {8}{3} a \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-4 a^{3} b +\frac {3}{2} a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (40 a^{4}-12 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{6}}}{d}\) | \(301\) |
| risch | \(-\frac {5 x \,a^{4}}{b^{6}}+\frac {3 x \,a^{2}}{2 b^{4}}+\frac {x}{8 b^{2}}+\frac {2 i a^{4} \left (a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{6} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}+\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{4 b^{3} d}-\frac {3 i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{4} d}+\frac {3 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{4} d}-\frac {2 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{b^{5} d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{4 b^{3} d}+\frac {2 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{b^{5} d}-\frac {5 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{6}}+\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{4}}+\frac {5 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{6}}-\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{4}}-\frac {\sin \left (4 d x +4 c \right )}{32 b^{2} d}+\frac {a \sin \left (3 d x +3 c \right )}{6 b^{3} d}\) | \(535\) |
1/d*(2*a^3/b^6*(a*b*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d *x+1/2*c)^2+a+b)+(5*a^2-4*b^2)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d* x+1/2*c)/((a-b)*(a+b))^(1/2)))-2/b^6*(((-4*a^3*b-3/2*a^2*b^2-1/8*b^4)*tan( 1/2*d*x+1/2*c)^7+(-12*a^3*b-3/2*a^2*b^2+8/3*a*b^3+7/8*b^4)*tan(1/2*d*x+1/2 *c)^5+(-12*a^3*b+3/2*a^2*b^2-7/8*b^4+8/3*a*b^3)*tan(1/2*d*x+1/2*c)^3+(-4*a ^3*b+3/2*a^2*b^2+1/8*b^4)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^4+1 /8*(40*a^4-12*a^2*b^2-b^4)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.36 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.08 \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\left [-\frac {3 \, {\left (40 \, a^{6} b - 52 \, a^{4} b^{3} + 11 \, a^{2} b^{5} + b^{7}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (40 \, a^{7} - 52 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + a b^{6}\right )} d x - 12 \, {\left (5 \, a^{6} - 4 \, a^{4} b^{2} + {\left (5 \, a^{5} b - 4 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (120 \, a^{6} b - 136 \, a^{4} b^{3} + 16 \, a^{2} b^{5} - 6 \, {\left (a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (20 \, a^{4} b^{3} - 23 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (60 \, a^{5} b^{2} - 73 \, a^{3} b^{4} + 13 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{6} - a b^{8}\right )} d\right )}}, -\frac {3 \, {\left (40 \, a^{6} b - 52 \, a^{4} b^{3} + 11 \, a^{2} b^{5} + b^{7}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (40 \, a^{7} - 52 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + a b^{6}\right )} d x - 24 \, {\left (5 \, a^{6} - 4 \, a^{4} b^{2} + {\left (5 \, a^{5} b - 4 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (120 \, a^{6} b - 136 \, a^{4} b^{3} + 16 \, a^{2} b^{5} - 6 \, {\left (a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (20 \, a^{4} b^{3} - 23 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (60 \, a^{5} b^{2} - 73 \, a^{3} b^{4} + 13 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{6} - a b^{8}\right )} d\right )}}\right ] \]
[-1/24*(3*(40*a^6*b - 52*a^4*b^3 + 11*a^2*b^5 + b^7)*d*x*cos(d*x + c) + 3* (40*a^7 - 52*a^5*b^2 + 11*a^3*b^4 + a*b^6)*d*x - 12*(5*a^6 - 4*a^4*b^2 + ( 5*a^5*b - 4*a^3*b^3)*cos(d*x + c))*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)) - (120*a^6*b - 136*a^4*b^3 + 16*a^2*b^5 - 6*(a^2*b^5 - b^7)*cos(d*x + c)^4 + 10*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (20*a^4*b^3 - 23*a^2*b^5 + 3* b^7)*cos(d*x + c)^2 + (60*a^5*b^2 - 73*a^3*b^4 + 13*a*b^6)*cos(d*x + c))*s in(d*x + c))/((a^2*b^7 - b^9)*d*cos(d*x + c) + (a^3*b^6 - a*b^8)*d), -1/24 *(3*(40*a^6*b - 52*a^4*b^3 + 11*a^2*b^5 + b^7)*d*x*cos(d*x + c) + 3*(40*a^ 7 - 52*a^5*b^2 + 11*a^3*b^4 + a*b^6)*d*x - 24*(5*a^6 - 4*a^4*b^2 + (5*a^5* b - 4*a^3*b^3)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/ (sqrt(a^2 - b^2)*sin(d*x + c))) - (120*a^6*b - 136*a^4*b^3 + 16*a^2*b^5 - 6*(a^2*b^5 - b^7)*cos(d*x + c)^4 + 10*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 - ( 20*a^4*b^3 - 23*a^2*b^5 + 3*b^7)*cos(d*x + c)^2 + (60*a^5*b^2 - 73*a^3*b^4 + 13*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^7 - b^9)*d*cos(d*x + c) + (a^3*b^6 - a*b^8)*d)]
Timed out. \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.33 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.78 \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {48 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\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 )} b^{5}} - \frac {3 \, {\left (40 \, a^{4} - 12 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {48 \, {\left (5 \, a^{5} - 4 \, a^{3} b^{2}\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 )}}{\sqrt {a^{2} - b^{2}} b^{6}} + \frac {2 \, {\left (96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 64 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 64 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, b^{3} \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 )}^{4} b^{5}}}{24 \, d} \]
1/24*(48*a^4*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d *x + 1/2*c)^2 + a + b)*b^5) - 3*(40*a^4 - 12*a^2*b^2 - b^4)*(d*x + c)/b^6 - 48*(5*a^5 - 4*a^3*b^2)*(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)))/(sqrt(a^2 - b^2)*b^6) + 2*(96*a^3*tan(1/2*d*x + 1/2*c)^7 + 36*a^2*b*t an(1/2*d*x + 1/2*c)^7 + 3*b^3*tan(1/2*d*x + 1/2*c)^7 + 288*a^3*tan(1/2*d*x + 1/2*c)^5 + 36*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 64*a*b^2*tan(1/2*d*x + 1/2 *c)^5 - 21*b^3*tan(1/2*d*x + 1/2*c)^5 + 288*a^3*tan(1/2*d*x + 1/2*c)^3 - 3 6*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 64*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 21*b^3* tan(1/2*d*x + 1/2*c)^3 + 96*a^3*tan(1/2*d*x + 1/2*c) - 36*a^2*b*tan(1/2*d* x + 1/2*c) - 3*b^3*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*b ^5))/d
Time = 5.74 (sec) , antiderivative size = 2971, normalized size of antiderivative = 12.54 \[ \int \frac {\cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]
(atan(((((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a ^7*b^4 + 3840*a^8*b^3 + 1280*a^9*b^2))/(2*b^10) - (((4*b^18 - 4*a*b^17 + 4 4*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^1 5 - (tan(c/2 + (d*x)/2)*(b^4*1i - a^4*40i + a^2*b^2*12i)*(128*a*b^14 - 256 *a^2*b^13 + 128*a^3*b^12))/(16*b^16))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8 *b^6))*(b^4*1i - a^4*40i + a^2*b^2*12i)*1i)/(8*b^6) + (((tan(c/2 + (d*x)/2 )*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 1 36*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3 + 128 0*a^9*b^2))/(2*b^10) + (((4*b^18 - 4*a*b^17 + 44*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 + (tan(c/2 + (d*x)/2)*(b^ 4*1i - a^4*40i + a^2*b^2*12i)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/ (16*b^16))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8*b^6))*(b^4*1i - a^4*40i + a^2*b^2*12i)*1i)/(8*b^6))/((12000*a^13*b - 8000*a^14 - 4*a^3*b^11 + 8*a^4* b^10 - 95*a^5*b^9 + 54*a^6*b^8 - 99*a^7*b^7 - 944*a^8*b^6 + 5240*a^9*b^5 + 440*a^10*b^4 - 15800*a^11*b^3 + 7200*a^12*b^2)/b^15 - (((tan(c/2 + (d*x)/ 2)*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3 + 12 80*a^9*b^2))/(2*b^10) - (((4*b^18 - 4*a*b^17 + 44*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 - (tan(c/2 + (d*x)/2)...