Integrand size = 33, antiderivative size = 372 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) x}{2 b^5}-\frac {a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
1/2*(2*A*b^2+(12*a^2+b^2)*C)*x/b^5-a*(6*A*b^6+a^4*b^2*(2*A-29*C)-5*a^2*b^4 *(A-4*C)+12*a^6*C)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b )^(5/2)/b^5/(a+b)^(5/2)/d-1/2*a*(a^2*b^2*(2*A-21*C)-b^4*(5*A-6*C)+12*a^4*C )*sin(d*x+c)/b^4/(a^2-b^2)^2/d+1/2*(a^2*b^2*(A-10*C)-b^4*(4*A-C)+6*a^4*C)* cos(d*x+c)*sin(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*(A*b^2+C*a^2)*cos(d*x+c)^3*sin (d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/2*(3*A*b^4-4*C*a^4+7*C*a^2*b^2) *cos(d*x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
Time = 4.25 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) (c+d x)+\frac {4 a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}-12 a b C \sin (c+d x)+\frac {2 a^3 b \left (A b^2+a^2 C\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {2 a^2 b \left (6 A b^4-7 a^4 C+a^2 b^2 (-3 A+10 C)\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+b^2 C \sin (2 (c+d x))}{4 b^5 d} \]
(2*(2*A*b^2 + (12*a^2 + b^2)*C)*(c + d*x) + (4*a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2] )/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 12*a*b*C*Sin[c + d*x] + (2*a^3*b *(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (2*a^2*b*(6*A*b^4 - 7*a^4*C + a^2*b^2*(-3*A + 10*C))*Sin[c + d*x])/((a - b )^2*(a + b)^2*(a + b*Cos[c + d*x])) + b^2*C*Sin[2*(c + d*x)])/(4*b^5*d)
Time = 2.14 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3042, 3527, 3042, 3526, 3042, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (-2 \left (2 C a^2+A b^2-b^2 C\right ) \cos ^2(c+d x)-2 a b (A+C) \cos (c+d x)+3 \left (C a^2+A b^2\right )\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-2 \left (2 C a^2+A b^2-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (C a^2+A b^2\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (2 \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right ) \cos ^2(c+d x)-a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \cos (c+d x)+2 \left (-4 C a^4+7 b^2 C a^2+3 A b^4\right )\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (-4 C a^4+7 b^2 C a^2+3 A b^4\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {2 \left (-a \left (12 C a^4+b^2 (2 A-21 C) a^2-b^4 (5 A-6 C)\right ) \cos ^2(c+d x)-b \left (2 C a^4-b^2 (A+4 C) a^2-b^4 (2 A+C)\right ) \cos (c+d x)+a \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )\right )}{a+b \cos (c+d x)}dx}{2 b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {-a \left (12 C a^4+b^2 (2 A-21 C) a^2-b^4 (5 A-6 C)\right ) \cos ^2(c+d x)-b \left (2 C a^4-b^2 (A+4 C) a^2-b^4 (2 A+C)\right ) \cos (c+d x)+a \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )}{a+b \cos (c+d x)}dx}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {-a \left (12 C a^4+b^2 (2 A-21 C) a^2-b^4 (5 A-6 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (2 C a^4-b^2 (A+4 C) a^2-b^4 (2 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \cos (c+d x) \left (a^2-b^2\right )^2+a b \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )}{a+b \cos (c+d x)}dx}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \sin (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2+a b \left (6 C a^4+b^2 (A-10 C) a^2-b^4 (4 A-C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \sin (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right )}{b}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \sin (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right )}{b}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \sin (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right )}{b}-\frac {2 a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \sin (c+d x)}{b d}}{b}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \sin (c+d x) \cos (c+d x)}{b d}+\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (C \left (12 a^2+b^2\right )+2 A b^2\right )}{b}-\frac {2 a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \sin (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\) |
-1/2*((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b *Cos[c + d*x])^2) - (-(((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*Cos[c + d*x]^2*S in[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))) - (((a^2*b^2*(A - 10* C) - b^4*(4*A - C) + 6*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(b*d) + ((((a^2 - b^2)^2*(2*A*b^2 + (12*a^2 + b^2)*C)*x)/b - (2*a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x) /2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d))/b - (a*(a^2*b^2*(2*A - 2 1*C) - b^4*(5*A - 6*C) + 12*a^4*C)*Sin[c + d*x])/(b*d))/b)/(b*(a^2 - b^2)) )/(2*b*(a^2 - b^2))
3.6.78.3.1 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_.) + (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_.) + (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])))
Time = 3.00 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b^{2}-A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}-C \,a^{3} b -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (2 A \,a^{2} b^{2}+A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}+C \,a^{3} b -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (2 A \,a^{4} b^{2}-5 A \,a^{2} b^{4}+6 A \,b^{6}+12 C \,a^{6}-29 C \,a^{4} b^{2}+20 C \,a^{2} b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}+\frac {\frac {2 \left (\left (-3 C a b -\frac {1}{2} b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 C a b +\frac {1}{2} b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}+12 a^{2} C +b^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(400\) |
| default | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b^{2}-A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}-C \,a^{3} b -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (2 A \,a^{2} b^{2}+A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}+C \,a^{3} b -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (2 A \,a^{4} b^{2}-5 A \,a^{2} b^{4}+6 A \,b^{6}+12 C \,a^{6}-29 C \,a^{4} b^{2}+20 C \,a^{2} b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}+\frac {\frac {2 \left (\left (-3 C a b -\frac {1}{2} b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 C a b +\frac {1}{2} b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}+12 a^{2} C +b^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(400\) |
| risch | \(\text {Expression too large to display}\) | \(1460\) |
1/d*(-2*a/b^5*((1/2*(2*A*a^2*b^2-A*a*b^3-6*A*b^4+6*C*a^4-C*a^3*b-10*C*a^2* b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*b*a*(2*A*a^2*b^2+A *a*b^3-6*A*b^4+6*C*a^4+C*a^3*b-10*C*a^2*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2 *c))/(tan(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d*x+1/2*c)^2+a+b)^2+1/2*(2*A*a^4*b^ 2-5*A*a^2*b^4+6*A*b^6+12*C*a^6-29*C*a^4*b^2+20*C*a^2*b^4)/(a^4-2*a^2*b^2+b ^4)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2 )))+2/b^5*(((-3*C*a*b-1/2*b^2*C)*tan(1/2*d*x+1/2*c)^3+(-3*C*a*b+1/2*b^2*C) *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+12*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. 732 vs. \(2 (353) = 706\).
Time = 0.41 (sec) , antiderivative size = 1535, normalized size of antiderivative = 4.13 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
[1/4*(2*(12*C*a^8*b^2 + (2*A - 35*C)*a^6*b^4 - 3*(2*A - 11*C)*a^4*b^6 + 3* (2*A - 3*C)*a^2*b^8 - (2*A + C)*b^10)*d*x*cos(d*x + c)^2 + 4*(12*C*a^9*b + (2*A - 35*C)*a^7*b^3 - 3*(2*A - 11*C)*a^5*b^5 + 3*(2*A - 3*C)*a^3*b^7 - ( 2*A + C)*a*b^9)*d*x*cos(d*x + c) + 2*(12*C*a^10 + (2*A - 35*C)*a^8*b^2 - 3 *(2*A - 11*C)*a^6*b^4 + 3*(2*A - 3*C)*a^4*b^6 - (2*A + C)*a^2*b^8)*d*x - ( 12*C*a^9 + (2*A - 29*C)*a^7*b^2 - 5*(A - 4*C)*a^5*b^4 + 6*A*a^3*b^6 + (12* C*a^7*b^2 + (2*A - 29*C)*a^5*b^4 - 5*(A - 4*C)*a^3*b^6 + 6*A*a*b^8)*cos(d* x + c)^2 + 2*(12*C*a^8*b + (2*A - 29*C)*a^6*b^3 - 5*(A - 4*C)*a^4*b^5 + 6* A*a^2*b^7)*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)) - 2*(1 2*C*a^9*b + (2*A - 33*C)*a^7*b^3 - (7*A - 27*C)*a^5*b^5 + (5*A - 6*C)*a^3* b^7 - (C*a^6*b^4 - 3*C*a^4*b^6 + 3*C*a^2*b^8 - C*b^10)*cos(d*x + c)^3 + 4* (C*a^7*b^3 - 3*C*a^5*b^5 + 3*C*a^3*b^7 - C*a*b^9)*cos(d*x + c)^2 + (18*C*a ^8*b^2 + (3*A - 50*C)*a^6*b^4 - (9*A - 43*C)*a^4*b^6 + (6*A - 11*C)*a^2*b^ 8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)* d*cos(d*x + c)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c) + (a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*d), 1/2*((12*C*a^8*b^ 2 + (2*A - 35*C)*a^6*b^4 - 3*(2*A - 11*C)*a^4*b^6 + 3*(2*A - 3*C)*a^2*b^8 - (2*A + C)*b^10)*d*x*cos(d*x + c)^2 + 2*(12*C*a^9*b + (2*A - 35*C)*a^7...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 2494 vs. \(2 (353) = 706\).
Time = 0.62 (sec) , antiderivative size = 2494, normalized size of antiderivative = 6.70 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
-1/2*(((2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 4*a*b^5 + 2*b^6)*sqrt(a^2 - b^2) *A*abs(a^4*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) + (12*a^6 - 6*a^5*b - 23*a^4 *b^2 + 10*a^3*b^3 + 10*a^2*b^4 - a*b^5 + b^6)*sqrt(a^2 - b^2)*C*abs(a^4*b^ 5 - 2*a^2*b^7 + b^9)*abs(-a + b) + (4*a^9*b^6 - 2*a^8*b^7 - 17*a^7*b^8 + 8 *a^6*b^9 + 30*a^5*b^10 - 12*a^4*b^11 - 25*a^3*b^12 + 8*a^2*b^13 + 8*a*b^14 - 2*b^15)*sqrt(a^2 - b^2)*A*abs(-a + b) + (24*a^11*b^4 - 12*a^10*b^5 - 10 0*a^9*b^6 + 47*a^8*b^7 + 158*a^7*b^8 - 68*a^6*b^9 - 111*a^5*b^10 + 42*a^4* b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14 - b^15)*sqrt(a^2 - b^2)*C*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/2*c)/sq rt((4*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 + sqrt(-16*(a^5*b^4 + a^4*b^5 - 2*a^3* b^6 - 2*a^2*b^7 + a*b^8 + b^9)*(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9) + 16*(a^5*b^4 - 2*a^3*b^6 + a*b^8)^2))/(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9))))/((a^4*b^5 - 2*a^2*b^7 + b^9)^2*(a ^2 - 2*a*b + b^2) + (a^7*b^4 - 2*a^6*b^5 - a^5*b^6 + 4*a^4*b^7 - a^3*b^8 - 2*a^2*b^9 + a*b^10)*abs(a^4*b^5 - 2*a^2*b^7 + b^9)) - (24*C*a^11*b^4 - 12 *C*a^10*b^5 + 4*A*a^9*b^6 - 100*C*a^9*b^6 - 2*A*a^8*b^7 + 47*C*a^8*b^7 - 1 7*A*a^7*b^8 + 158*C*a^7*b^8 + 8*A*a^6*b^9 - 68*C*a^6*b^9 + 30*A*a^5*b^10 - 111*C*a^5*b^10 - 12*A*a^4*b^11 + 42*C*a^4*b^11 - 25*A*a^3*b^12 + 28*C*a^3 *b^12 + 8*A*a^2*b^13 - 8*C*a^2*b^13 + 8*A*a*b^14 + C*a*b^14 - 2*A*b^15 - C *b^15 - 12*C*a^6*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + 6*C*a^5*b*abs(a^4*b^5...
Time = 15.13 (sec) , antiderivative size = 10483, normalized size of antiderivative = 28.18 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
(a*atan(((a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a ^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8 + 32*A^2*a^7*b ^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 21*C^2*a^2*b^12 - 4 0*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872* C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 110 4*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 8*A*C*a*b^13 + 36*A*C*a^ 2*b^12 - 64*A*C*a^3*b^11 + 104*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^ 6*b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*C*a^9*b^5 - 376*A*C*a^10 *b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3 *a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) + (a*(-(a + b)^5* (a - b)^5)^(1/2)*((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*A*a^8*b ^13 - 8*A*a^9*b^12 + 28*C*a^2*b^19 - 80*C*a^3*b^18 - 120*C*a^4*b^17 + 276* C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*C*a^9* b^12 + 24*C*a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3* a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - ( 4*a*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 20*C*a^2*b^4 - 29*C*a^4*b^2)*(8*a*b^19 - 8*a^2 *b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*...