Integrand size = 41, antiderivative size = 649 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {\left (2 A b^2-8 a b B+20 a^2 C+b^2 C\right ) x}{2 b^6}+\frac {a \left (8 A b^8+8 a^7 b B-28 a^5 b^3 B+35 a^3 b^5 B-20 a b^7 B-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-20 a^8 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^6 \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {\left (24 a^6 b B-68 a^4 b^3 B+65 a^2 b^5 B-6 b^7 B-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-2 a b^6 (13 A-12 C)-60 a^7 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}-\frac {\left (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-a^4 b^2 (A-27 C)+a^2 b^4 (2 A-23 C)-b^6 (6 A-C)-10 a^6 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^4(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (4 A b^4+2 a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (12 A b^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:
1/2*(2*A*b^2-8*B*a*b+20*C*a^2+C*b^2)*x/b^6+a*(8*A*b^8+8*a^7*b*B-28*a^5*b^3 *B+35*a^3*b^5*B-20*a*b^7*B-a^6*b^2*(2*A-69*C)+7*a^4*b^4*(A-12*C)-8*a^2*b^6 *(A-5*C)-20*a^8*C)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b )^(1/2)/b^6/(a+b)^(1/2)/(a^2-b^2)^3/d+1/6*(24*a^6*b*B-68*a^4*b^3*B+65*a^2* b^5*B-6*b^7*B-a^5*b^2*(6*A-167*C)+a^3*b^4*(17*A-146*C)-2*a*b^6*(13*A-12*C) -60*a^7*C)*sin(d*x+c)/b^5/(a^2-b^2)^3/d-1/2*(4*B*a^5*b-11*B*a^3*b^3+12*B*a *b^5-a^4*b^2*(A-27*C)+a^2*b^4*(2*A-23*C)-b^6*(6*A-C)-10*a^6*C)*cos(d*x+c)* sin(d*x+c)/b^4/(a^2-b^2)^3/d-1/3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^4*sin(d*x+ c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^3+1/6*(4*A*b^4+2*B*a^3*b-7*B*a*b^3-5*a^4 *C+a^2*b^2*(A+10*C))*cos(d*x+c)^3*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d* x+c))^2-1/6*(12*A*b^6-8*B*a^5*b+20*B*a^3*b^3-27*B*a*b^5+a^4*b^2*(2*A-53*C) +20*a^6*C+a^2*b^4*(A+48*C))*cos(d*x+c)^2*sin(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b *cos(d*x+c))
Result contains complex when optimal does not.
Time = 11.69 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {\left (2 A b^2-8 a b B+20 a^2 C+b^2 C\right ) (c+d x)}{2 b^6 d}+\frac {a \left (2 a^6 A b^2-7 a^4 A b^4+8 a^2 A b^6-8 A b^8-8 a^7 b B+28 a^5 b^3 B-35 a^3 b^5 B+20 a b^7 B+20 a^8 C-69 a^6 b^2 C+84 a^4 b^4 C-40 a^2 b^6 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{b^6 \left (a^2-b^2\right )^3 \sqrt {-a^2+b^2} d}+\frac {(-b B+4 a C) \left (-\frac {i \cos (c+d x)}{2 b^5}-\frac {\sin (c+d x)}{2 b^5}\right )}{d}+\frac {(-b B+4 a C) \left (\frac {i \cos (c+d x)}{2 b^5}-\frac {\sin (c+d x)}{2 b^5}\right )}{d}+\frac {a^4 A b^2 \sin (c+d x)-a^5 b B \sin (c+d x)+a^6 C \sin (c+d x)}{3 b^5 \left (-a^2+b^2\right ) d (a+b \cos (c+d x))^3}+\frac {7 a^5 A b^2 \sin (c+d x)-12 a^3 A b^4 \sin (c+d x)-10 a^6 b B \sin (c+d x)+15 a^4 b^3 B \sin (c+d x)+13 a^7 C \sin (c+d x)-18 a^5 b^2 C \sin (c+d x)}{6 b^5 \left (-a^2+b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {11 a^6 A b^2 \sin (c+d x)-32 a^4 A b^4 \sin (c+d x)+36 a^2 A b^6 \sin (c+d x)-26 a^7 b B \sin (c+d x)+71 a^5 b^3 B \sin (c+d x)-60 a^3 b^5 B \sin (c+d x)+47 a^8 C \sin (c+d x)-122 a^6 b^2 C \sin (c+d x)+90 a^4 b^4 C \sin (c+d x)}{6 b^5 \left (-a^2+b^2\right )^3 d (a+b \cos (c+d x))}+\frac {C \sin (2 (c+d x))}{4 b^4 d} \] Input:
Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b* Cos[c + d*x])^4,x]
Output:
((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*(c + d*x))/(2*b^6*d) + (a*(2*a^6*A *b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 28*a^5*b^3*B - 35 *a^3*b^5*B + 20*a*b^7*B + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2* b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(b^6*(a^2 - b ^2)^3*Sqrt[-a^2 + b^2]*d) + ((-(b*B) + 4*a*C)*(((-1/2*I)*Cos[c + d*x])/b^5 - Sin[c + d*x]/(2*b^5)))/d + ((-(b*B) + 4*a*C)*(((I/2)*Cos[c + d*x])/b^5 - Sin[c + d*x]/(2*b^5)))/d + (a^4*A*b^2*Sin[c + d*x] - a^5*b*B*Sin[c + d*x ] + a^6*C*Sin[c + d*x])/(3*b^5*(-a^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (7 *a^5*A*b^2*Sin[c + d*x] - 12*a^3*A*b^4*Sin[c + d*x] - 10*a^6*b*B*Sin[c + d *x] + 15*a^4*b^3*B*Sin[c + d*x] + 13*a^7*C*Sin[c + d*x] - 18*a^5*b^2*C*Sin [c + d*x])/(6*b^5*(-a^2 + b^2)^2*d*(a + b*Cos[c + d*x])^2) + (11*a^6*A*b^2 *Sin[c + d*x] - 32*a^4*A*b^4*Sin[c + d*x] + 36*a^2*A*b^6*Sin[c + d*x] - 26 *a^7*b*B*Sin[c + d*x] + 71*a^5*b^3*B*Sin[c + d*x] - 60*a^3*b^5*B*Sin[c + d *x] + 47*a^8*C*Sin[c + d*x] - 122*a^6*b^2*C*Sin[c + d*x] + 90*a^4*b^4*C*Si n[c + d*x])/(6*b^5*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(4*b^4*d)
Time = 3.88 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 3526, 3042, 3526, 3042, 3526, 3042, 3528, 27, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {\cos ^3(c+d x) \left (-\left (\left (5 C a^2-2 b B a+2 A b^2-3 b^2 C\right ) \cos ^2(c+d x)\right )+3 b (b B-a (A+C)) \cos (c+d x)+4 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\left (-5 C a^2+2 b B a-2 A b^2+3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+4 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos ^2(c+d x) \left (-2 \left (-10 C a^4+4 b B a^3-b^2 (A-18 C) a^2-9 b^3 B a+3 b^4 (2 A-C)\right ) \cos ^2(c+d x)+2 b \left (C a^3+2 b B a^2-b^2 (5 A+6 C) a+3 b^3 B\right ) \cos (c+d x)+3 \left (-5 C a^4+2 b B a^3+b^2 (A+10 C) a^2-7 b^3 B a+4 A b^4\right )\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-2 \left (-10 C a^4+4 b B a^3-b^2 (A-18 C) a^2-9 b^3 B a+3 b^4 (2 A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (C a^3+2 b B a^2-b^2 (5 A+6 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-5 C a^4+2 b B a^3+b^2 (A+10 C) a^2-7 b^3 B a+4 A b^4\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {\cos (c+d x) \left (6 \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right ) \cos ^2(c+d x)+b \left (-5 C a^5+2 b B a^4-b^2 (5 A-8 C) a^3+7 b^3 B a^2-2 b^4 (5 A+9 C) a+6 b^5 B\right ) \cos (c+d x)+2 \left (20 C a^6-8 b B a^5+b^2 (2 A-53 C) a^4+20 b^3 B a^3+b^4 (A+48 C) a^2-27 b^5 B a+12 A b^6\right )\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (6 \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-5 C a^5+2 b B a^4-b^2 (5 A-8 C) a^3+7 b^3 B a^2-2 b^4 (5 A+9 C) a+6 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (20 C a^6-8 b B a^5+b^2 (2 A-53 C) a^4+20 b^3 B a^3+b^4 (A+48 C) a^2-27 b^5 B a+12 A b^6\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {2 \left (-\left (\left (-60 C a^7+24 b B a^6-b^2 (6 A-167 C) a^5-68 b^3 B a^4+b^4 (17 A-146 C) a^3+65 b^5 B a^2-2 b^6 (13 A-12 C) a-6 b^7 B\right ) \cos ^2(c+d x)\right )-b \left (-10 C a^6+4 b B a^5-b^2 (A-25 C) a^4-7 b^3 B a^3-b^4 (8 A+27 C) a^2+18 b^5 B a-3 b^6 (2 A+C)\right ) \cos (c+d x)+3 a \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right )\right )}{a+b \cos (c+d x)}dx}{2 b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {-\left (\left (-60 C a^7+24 b B a^6-b^2 (6 A-167 C) a^5-68 b^3 B a^4+b^4 (17 A-146 C) a^3+65 b^5 B a^2-2 b^6 (13 A-12 C) a-6 b^7 B\right ) \cos ^2(c+d x)\right )-b \left (-10 C a^6+4 b B a^5-b^2 (A-25 C) a^4-7 b^3 B a^3-b^4 (8 A+27 C) a^2+18 b^5 B a-3 b^6 (2 A+C)\right ) \cos (c+d x)+3 a \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right )}{a+b \cos (c+d x)}dx}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {\left (60 C a^7-24 b B a^6+b^2 (6 A-167 C) a^5+68 b^3 B a^4-b^4 (17 A-146 C) a^3-65 b^5 B a^2+2 b^6 (13 A-12 C) a+6 b^7 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-10 C a^6+4 b B a^5-b^2 (A-25 C) a^4-7 b^3 B a^3-b^4 (8 A+27 C) a^2+18 b^5 B a-3 b^6 (2 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\int \frac {3 \left (a b \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right )-\left (a^2-b^2\right )^3 \left (20 C a^2-8 b B a+2 A b^2+b^2 C\right ) \cos (c+d x)\right )}{a+b \cos (c+d x)}dx}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {3 \int \frac {a b \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right )-\left (a^2-b^2\right )^3 \left (20 C a^2-8 b B a+2 A b^2+b^2 C\right ) \cos (c+d x)}{a+b \cos (c+d x)}dx}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {3 \int \frac {a b \left (-10 C a^6+4 b B a^5-b^2 (A-27 C) a^4-11 b^3 B a^3+b^4 (2 A-23 C) a^2+12 b^5 B a-b^6 (6 A-C)\right )-\left (a^2-b^2\right )^3 \left (20 C a^2-8 b B a+2 A b^2+b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {3 \left (-\frac {a \left (-20 a^8 C+8 a^7 b B-a^6 b^2 (2 A-69 C)-28 a^5 b^3 B+7 a^4 b^4 (A-12 C)+35 a^3 b^5 B-8 a^2 b^6 (A-5 C)-20 a b^7 B+8 A b^8\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}-\frac {x \left (a^2-b^2\right )^3 \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{b}\right )}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {3 \left (-\frac {a \left (-20 a^8 C+8 a^7 b B-a^6 b^2 (2 A-69 C)-28 a^5 b^3 B+7 a^4 b^4 (A-12 C)+35 a^3 b^5 B-8 a^2 b^6 (A-5 C)-20 a b^7 B+8 A b^8\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {x \left (a^2-b^2\right )^3 \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{b}\right )}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {3 \left (-\frac {2 a \left (-20 a^8 C+8 a^7 b B-a^6 b^2 (2 A-69 C)-28 a^5 b^3 B+7 a^4 b^4 (A-12 C)+35 a^3 b^5 B-8 a^2 b^6 (A-5 C)-20 a b^7 B+8 A b^8\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}-\frac {x \left (a^2-b^2\right )^3 \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{b}\right )}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {-\frac {\sin (c+d x) \cos ^3(c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\sin (c+d x) \cos ^2(c+d x) \left (20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right )}{b d}+\frac {\frac {3 \left (-\frac {x \left (a^2-b^2\right )^3 \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{b}-\frac {2 a \left (-20 a^8 C+8 a^7 b B-a^6 b^2 (2 A-69 C)-28 a^5 b^3 B+7 a^4 b^4 (A-12 C)+35 a^3 b^5 B-8 a^2 b^6 (A-5 C)-20 a b^7 B+8 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}\right )}{b}-\frac {\sin (c+d x) \left (-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]
Output:
-1/3*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^4*Sin[c + d*x])/(b*(a^2 - b^2)* d*(a + b*Cos[c + d*x])^3) - (-1/2*((4*A*b^4 + 2*a^3*b*B - 7*a*b^3*B - 5*a^ 4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (-(((12*A*b^6 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^ 5*B + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c + d*x]^2 *Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))) - ((3*(4*a^5*b*B - 11*a^3*b^3*B + 12*a*b^5*B - a^4*b^2*(A - 27*C) + a^2*b^4*(2*A - 23*C) - b^ 6*(6*A - C) - 10*a^6*C)*Cos[c + d*x]*Sin[c + d*x])/(b*d) + ((3*(-(((a^2 - b^2)^3*(2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/b) - (2*a*(8*A*b^8 + 8*a^ 7*b*B - 28*a^5*b^3*B + 35*a^3*b^5*B - 20*a*b^7*B - a^6*b^2*(2*A - 69*C) + 7*a^4*b^4*(A - 12*C) - 8*a^2*b^6*(A - 5*C) - 20*a^8*C)*ArcTan[(Sqrt[a - b] *Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)))/b - ((24* a^6*b*B - 68*a^4*b^3*B + 65*a^2*b^5*B - 6*b^7*B - a^5*b^2*(6*A - 167*C) + a^3*b^4*(17*A - 146*C) - 2*a*b^6*(13*A - 12*C) - 60*a^7*C)*Sin[c + d*x])/( b*d))/b)/(b*(a^2 - b^2)))/(2*b*(a^2 - b^2)))/(3*b*(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_)])/((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_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 4.32 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{4} b^{2}-A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+4 A a \,b^{5}+12 A \,b^{6}-6 B \,a^{5} b +2 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}-5 B \,a^{2} b^{4}-20 B a \,b^{5}+12 a^{6} C -3 C \,a^{5} b -34 a^{4} b^{2} C +6 C \,a^{3} b^{3}+30 a^{2} C \,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 (3 A \,a^{4} b^{2}-11 a^{2} A \,b^{4}+18 A \,b^{6}-9 B \,a^{5} b +29 B \,a^{3} b^{3}-30 B a \,b^{5}+18 a^{6} C -53 a^{4} b^{2} C +45 a^{2} C \,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 (2 A \,a^{4} b^{2}+A \,a^{3} b^{3}-6 a^{2} A \,b^{4}-4 A a \,b^{5}+12 A \,b^{6}-6 B \,a^{5} b -2 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}+5 B \,a^{2} b^{4}-20 B a \,b^{5}+12 a^{6} C +3 C \,a^{5} b -34 a^{4} b^{2} C -6 C \,a^{3} b^{3}+30 a^{2} C \,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 (2 A \,a^{6} b^{2}-7 a^{4} A \,b^{4}+8 a^{2} A \,b^{6}-8 A \,b^{8}-8 a^{7} b B +28 a^{5} b^{3} B -35 a^{3} b^{5} B +20 a \,b^{7} B +20 a^{8} C -69 a^{6} b^{2} C +84 a^{4} b^{4} C -40 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 )}{b^{6}}+\frac {\frac {2 \left (\left (B \,b^{2}-4 a b C -\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (B \,b^{2}-4 a b C +\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (2 A \,b^{2}-8 B a b +20 a^{2} C +C \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6}}}{d}\) | \(767\) |
| default | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{4} b^{2}-A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+4 A a \,b^{5}+12 A \,b^{6}-6 B \,a^{5} b +2 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}-5 B \,a^{2} b^{4}-20 B a \,b^{5}+12 a^{6} C -3 C \,a^{5} b -34 a^{4} b^{2} C +6 C \,a^{3} b^{3}+30 a^{2} C \,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 (3 A \,a^{4} b^{2}-11 a^{2} A \,b^{4}+18 A \,b^{6}-9 B \,a^{5} b +29 B \,a^{3} b^{3}-30 B a \,b^{5}+18 a^{6} C -53 a^{4} b^{2} C +45 a^{2} C \,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 (2 A \,a^{4} b^{2}+A \,a^{3} b^{3}-6 a^{2} A \,b^{4}-4 A a \,b^{5}+12 A \,b^{6}-6 B \,a^{5} b -2 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}+5 B \,a^{2} b^{4}-20 B a \,b^{5}+12 a^{6} C +3 C \,a^{5} b -34 a^{4} b^{2} C -6 C \,a^{3} b^{3}+30 a^{2} C \,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 (2 A \,a^{6} b^{2}-7 a^{4} A \,b^{4}+8 a^{2} A \,b^{6}-8 A \,b^{8}-8 a^{7} b B +28 a^{5} b^{3} B -35 a^{3} b^{5} B +20 a \,b^{7} B +20 a^{8} C -69 a^{6} b^{2} C +84 a^{4} b^{4} C -40 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 )}{b^{6}}+\frac {\frac {2 \left (\left (B \,b^{2}-4 a b C -\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (B \,b^{2}-4 a b C +\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (2 A \,b^{2}-8 B a b +20 a^{2} C +C \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6}}}{d}\) | \(767\) |
| risch | \(\text {Expression too large to display}\) | \(3290\) |
Input:
int(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x,meth od=_RETURNVERBOSE)
Output:
1/d*(-2*a/b^6*((1/2*(2*A*a^4*b^2-A*a^3*b^3-6*A*a^2*b^4+4*A*a*b^5+12*A*b^6- 6*B*a^5*b+2*B*a^4*b^2+18*B*a^3*b^3-5*B*a^2*b^4-20*B*a*b^5+12*C*a^6-3*C*a^5 *b-34*C*a^4*b^2+6*C*a^3*b^3+30*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*(3*A*a^4*b^2-11*A*a^2*b^4+18*A*b^6-9*B*a^5*b+ 29*B*a^3*b^3-30*B*a*b^5+18*C*a^6-53*C*a^4*b^2+45*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*(2*A*a^4*b^2+A*a^3*b^3-6*A* a^2*b^4-4*A*a*b^5+12*A*b^6-6*B*a^5*b-2*B*a^4*b^2+18*B*a^3*b^3+5*B*a^2*b^4- 20*B*a*b^5+12*C*a^6+3*C*a^5*b-34*C*a^4*b^2-6*C*a^3*b^3+30*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*(2*A*a^6*b^2-7*A*a^4*b^4+8*A*a^2*b^6-8* A*b^8-8*B*a^7*b+28*B*a^5*b^3-35*B*a^3*b^5+20*B*a*b^7+20*C*a^8-69*C*a^6*b^2 +84*C*a^4*b^4-40*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)))+2/b^6*(((B*b^2-4 *a*b*C-1/2*C*b^2)*tan(1/2*d*x+1/2*c)^3+(B*b^2-4*a*b*C+1/2*C*b^2)*tan(1/2*d *x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(2*A*b^2-8*B*a*b+20*C*a^2+C*b^2) *arctan(tan(1/2*d*x+1/2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (627) = 1254\).
Time = 0.50 (sec) , antiderivative size = 3385, normalized size of antiderivative = 5.22 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))* *4,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(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. 1436 vs. \(2 (627) = 1254\).
Time = 0.24 (sec) , antiderivative size = 1436, normalized size of antiderivative = 2.21 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="giac")
Output:
1/6*(6*(20*C*a^9 - 8*B*a^8*b + 2*A*a^7*b^2 - 69*C*a^7*b^2 + 28*B*a^6*b^3 - 7*A*a^5*b^4 + 84*C*a^5*b^4 - 35*B*a^4*b^5 + 8*A*a^3*b^6 - 40*C*a^3*b^6 + 20*B*a^2*b^7 - 8*A*a*b^8)*(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^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*sqrt(a^2 - b^2)) - 2*(36* C*a^10*tan(1/2*d*x + 1/2*c)^5 - 18*B*a^9*b*tan(1/2*d*x + 1/2*c)^5 - 81*C*a ^9*b*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 + 42*B*a^ 8*b^2*tan(1/2*d*x + 1/2*c)^5 - 48*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 15*A* a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 213 *C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 1 17*B*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 48*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 45*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^ 5 - 162*C*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^4*b^6*tan(1/2*d*x + 1/2*c )^5 + 105*B*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 90*C*a^4*b^6*tan(1/2*d*x + 1/ 2*c)^5 - 60*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^10*tan(1/2*d*x + 1 /2*c)^3 - 36*B*a^9*b*tan(1/2*d*x + 1/2*c)^3 + 12*A*a^8*b^2*tan(1/2*d*x + 1 /2*c)^3 - 284*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 + 152*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^3 - 56*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 392*C*a^6*b^4*tan(1/2* d*x + 1/2*c)^3 - 236*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 + 116*A*a^4*b^6*t...
Time = 12.71 (sec) , antiderivative size = 21924, normalized size of antiderivative = 33.78 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^4,x)
Output:
(atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^18 + 800*C^2*a^18 + C^2*b^18 - 8*A ^2*a*b^17 - 2*C^2*a*b^17 - 800*C^2*a^17*b + 44*A^2*a^2*b^16 + 48*A^2*a^3*b ^15 - 92*A^2*a^4*b^14 - 120*A^2*a^5*b^13 + 156*A^2*a^6*b^12 + 160*A^2*a^7* b^11 - 164*A^2*a^8*b^10 - 120*A^2*a^9*b^9 + 117*A^2*a^10*b^8 + 48*A^2*a^11 *b^7 - 48*A^2*a^12*b^6 - 8*A^2*a^13*b^5 + 8*A^2*a^14*b^4 + 64*B^2*a^2*b^16 - 128*B^2*a^3*b^15 + 80*B^2*a^4*b^14 + 768*B^2*a^5*b^13 - 824*B^2*a^6*b^1 2 - 1920*B^2*a^7*b^11 + 2025*B^2*a^8*b^10 + 2560*B^2*a^9*b^9 - 2600*B^2*a^ 10*b^8 - 1920*B^2*a^11*b^7 + 1920*B^2*a^12*b^6 + 768*B^2*a^13*b^5 - 768*B^ 2*a^14*b^4 - 128*B^2*a^15*b^3 + 128*B^2*a^16*b^2 + 35*C^2*a^2*b^16 - 68*C^ 2*a^3*b^15 + 209*C^2*a^4*b^14 - 350*C^2*a^5*b^13 - 45*C^2*a^6*b^12 + 3640* C^2*a^7*b^11 - 3325*C^2*a^8*b^10 - 10430*C^2*a^9*b^9 + 10385*C^2*a^10*b^8 + 14812*C^2*a^11*b^7 - 14837*C^2*a^12*b^6 - 11522*C^2*a^13*b^5 + 11522*C^2 *a^14*b^4 + 4720*C^2*a^15*b^3 - 4720*C^2*a^16*b^2 + 4*A*C*b^18 - 32*A*B*a* b^17 - 8*A*C*a*b^17 - 16*B*C*a*b^17 - 640*B*C*a^17*b + 64*A*B*a^2*b^16 - 1 60*A*B*a^3*b^15 - 384*A*B*a^4*b^14 + 592*A*B*a^5*b^13 + 960*A*B*a^6*b^12 - 1128*A*B*a^7*b^11 - 1280*A*B*a^8*b^10 + 1306*A*B*a^9*b^9 + 960*A*B*a^10*b ^8 - 948*A*B*a^11*b^7 - 384*A*B*a^12*b^6 + 384*A*B*a^13*b^5 + 64*A*B*a^14* b^4 - 64*A*B*a^15*b^3 + 60*A*C*a^2*b^16 - 112*A*C*a^3*b^15 + 276*A*C*a^4*b ^14 + 840*A*C*a^5*b^13 - 1284*A*C*a^6*b^12 - 2240*A*C*a^7*b^11 + 2588*A*C* a^8*b^10 + 3080*A*C*a^9*b^9 - 3124*A*C*a^10*b^8 - 2352*A*C*a^11*b^7 + 2...
Time = 0.20 (sec) , antiderivative size = 5074, normalized size of antiderivative = 7.82 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x)
Output:
( - 120*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/s qrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**9*b**3*c + 36*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**5 + 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)**2*a**7*b**5*c - 126*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - t an((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**6*b* *7 - 504*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**5*b**7*c + 162*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**4*b**9 + 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)**2*a**3*b**9*c - 72*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**2*b **11 + 360*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**11*b*c - 108*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 **10*b**3 - 1122*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**9*b**3*c + 342*sqrt(a**2 - b**2 )*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos...