Integrand size = 20, antiderivative size = 210 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {3 a b \left (a^4-6 a^2 b^2+b^4\right ) x}{4 \left (a^2+b^2\right )^4}-\frac {b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {3 a^2 b^2 \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac {a b \left (5 a^2-3 b^2\right ) \cos (x) \sin (x)}{4 \left (a^2+b^2\right )^3}-\frac {a b \cos ^3(x) \sin (x)}{2 \left (a^2+b^2\right )^2}-\frac {2 a^2 b^2 \sin ^2(x)}{\left (a^2+b^2\right )^3}+\frac {a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
-3/4*a*b*(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4-1/4*b^2*cos(x)^4/(a^2+b^2)^2-3* a^2*b^2*(a^2-b^2)*ln(a*cos(x)+b*sin(x))/(a^2+b^2)^4+1/4*a*b*(5*a^2-3*b^2)* cos(x)*sin(x)/(a^2+b^2)^3-1/2*a*b*cos(x)^3*sin(x)/(a^2+b^2)^2-2*a^2*b^2*si n(x)^2/(a^2+b^2)^3+1/4*a^2*sin(x)^4/(a^2+b^2)^2-a^2*b^3*sin(x)/(a^2+b^2)^3 /(a*cos(x)+b*sin(x))
Result contains complex when optimal does not.
Time = 3.44 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.95 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {-12 a b \left (a^2-3 b^2\right ) \left (3 a^2-b^2\right ) x+6 i \left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) x-6 i \left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \arctan (\tan (x))-4 \left (a^2+b^2\right ) \left (a^4-6 a^2 b^2+b^4\right ) \cos (2 x)+\left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \cos (4 x)+3 \left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+\frac {2 b \left (a^2+b^2\right ) \left (3 a^4-10 a^2 b^2+3 b^4\right ) \sin (x)}{a \cos (x)+b \sin (x)}+\frac {3 \left (a^2+b^2\right )^2 \left (a \cos (x) \left (-2 i (a+i b)^2 x+\left (-a^2+b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )\right )+b \left (2 (a+i b) (a (-1-i x)+b (i+x))+\left (-a^2+b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )\right ) \sin (x)+2 i \left (a^2-b^2\right ) \arctan (\tan (x)) (a \cos (x)+b \sin (x))\right )}{a \cos (x)+b \sin (x)}+16 a b \left (a^4-b^4\right ) \sin (2 x)-2 a b \left (a^2+b^2\right )^2 \sin (4 x)}{32 \left (a^2+b^2\right )^4} \]
(-12*a*b*(a^2 - 3*b^2)*(3*a^2 - b^2)*x + (6*I)*(a^6 - 15*a^4*b^2 + 15*a^2* b^4 - b^6)*x - (6*I)*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*ArcTan[Tan[x]] - 4*(a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Cos[2*x] + (a^2 - b^2)*(a^2 + b^2) ^2*Cos[4*x] + 3*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Log[(a*Cos[x] + b*Si n[x])^2] + (2*b*(a^2 + b^2)*(3*a^4 - 10*a^2*b^2 + 3*b^4)*Sin[x])/(a*Cos[x] + b*Sin[x]) + (3*(a^2 + b^2)^2*(a*Cos[x]*((-2*I)*(a + I*b)^2*x + (-a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2]) + b*(2*(a + I*b)*(a*(-1 - I*x) + b*(I + x)) + (-a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2])*Sin[x] + (2*I)*(a^2 - b^ 2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x])))/(a*Cos[x] + b*Sin[x]) + 16*a*b*( a^4 - b^4)*Sin[2*x] - 2*a*b*(a^2 + b^2)^2*Sin[4*x])/(32*(a^2 + b^2)^4)
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.
| \(\displaystyle \int \frac {\sin ^3(x) \cos ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^3 \cos (x)^3}{(a \cos (x)+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {b \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos ^2(x) \sin ^3(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^3 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \left (\frac {b \int \cos ^2(x) \sin ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin ^3(x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \cos ^3(x) \sin (x)dx}{a^2+b^2}+\frac {a \int \cos ^2(x) \sin ^2(x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \cos (x)^3 \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)^3dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \left (\frac {a \int \sin ^3(x)d\sin (x)}{a^2+b^2}+\frac {b \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \cos (x)^3 \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \left (\frac {b \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \cos (x)^3 \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {b \left (-\frac {b \int \cos ^3(x)d\cos (x)}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {b \left (\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {a \left (\frac {b \left (\frac {1}{4} \int \cos ^2(x)dx-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {a \left (\frac {1}{4} \int \cos ^2(x)dx-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {b \left (\frac {1}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {a \left (\frac {1}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (\frac {b \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {a \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin ^2(x)dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \cos ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {b \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {b \left (\frac {b \int \cos ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \int \sin ^2(x)dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {a \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {b \left (\frac {b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (\frac {a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
3.3.92.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_. ) + (d_.)*(x_)]), x_Symbol] :> Simp[b*(x/(a^2 + b^2)), x] - Simp[a/(a^2 + b ^2) Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x ]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_. ) + (d_.)*(x_)]), x_Symbol] :> Simp[a*(x/(a^2 + b^2)), x] + Simp[b/(a^2 + b ^2) Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x ]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[a*(b/(a^ 2 + b^2)) Int[Cos[c + d*x]^(m - 1)*(Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Sim p[b/(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x], x] + (Simp[a/(a^2 + b^2) Int[Cos[c + d*x]^( m - 1)*Sin[c + d*x]^n*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x], x] - S imp[a*(b/(a^2 + b^2)) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1)*(a*Co s[c + d*x] + b*Sin[c + d*x])^p, x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^ 2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && ILtQ[p, 0]
Time = 1.13 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(\frac {a^{3} b^{2}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (x \right )\right )}-\frac {3 a^{2} b^{2} \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\frac {\left (\frac {1}{2} a^{3} b^{3}-\frac {3}{4} a \,b^{5}+\frac {5}{4} a^{5} b \right ) \tan \left (x \right )^{3}+\left (-\frac {1}{2} a^{6}+a^{4} b^{2}+\frac {3}{2} a^{2} b^{4}\right ) \tan \left (x \right )^{2}+\left (\frac {3}{4} a^{5} b -\frac {1}{2} a^{3} b^{3}-\frac {5}{4} a \,b^{5}\right ) \tan \left (x \right )-\frac {a^{6}}{4}+\frac {5 a^{4} b^{2}}{4}+\frac {5 a^{2} b^{4}}{4}-\frac {b^{6}}{4}}{\left (1+\tan \left (x \right )^{2}\right )^{2}}+\frac {3 a b \left (\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan \left (x \right )^{2}\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{4}}\) | \(232\) |
| parallelrisch | \(\frac {-192 b^{2} \left (a^{2}-b^{2}+\cos \left (2 x \right ) \left (a^{2}+b^{2}\right )\right ) \left (a -b \right ) a^{2} \left (a +b \right ) \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+192 b^{2} \left (a^{2}-b^{2}+\cos \left (2 x \right ) \left (a^{2}+b^{2}\right )\right ) \left (a -b \right ) a^{2} \left (a +b \right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-\left (a^{2}+b^{2}\right ) \left (48 x \,a^{5} b -288 x \,a^{3} b^{3}+48 a \,b^{5} x +a^{6}-15 a^{4} b^{2}+159 a^{2} b^{4}-17 b^{6}\right ) \cos \left (2 x \right )-2 \left (a^{4}-10 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )^{2} \cos \left (4 x \right )+\left (a -b \right ) \left (a +b \right ) \left (a^{2}+b^{2}\right )^{3} \cos \left (6 x \right )+\left (30 a^{7} b -102 a^{5} b^{3}-102 a^{3} b^{5}+30 b^{7} a \right ) \sin \left (2 x \right )+12 b a \left (a -b \right ) \left (a +b \right ) \left (a^{2}+b^{2}\right )^{2} \sin \left (4 x \right )-2 b a \left (a^{2}+b^{2}\right )^{3} \sin \left (6 x \right )-48 x \,a^{7} b +336 x \,a^{5} b^{3}-336 x \,a^{3} b^{5}+48 a \,b^{7} x +2 a^{8}-32 a^{6} b^{2}+108 a^{4} b^{4}+128 a^{2} b^{6}-14 b^{8}}{64 \left (a^{2}-b^{2}+\cos \left (2 x \right ) \left (a^{2}+b^{2}\right )\right ) \left (a^{2}+b^{2}\right )^{4}}\) | \(391\) |
| risch | \(\frac {3 x a b}{4 \left (4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}\right )}+\frac {{\mathrm e}^{4 i x}}{-128 i b a +64 a^{2}-64 b^{2}}-\frac {i {\mathrm e}^{2 i x} b}{16 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right )}-\frac {{\mathrm e}^{2 i x} a}{16 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right )}+\frac {i {\mathrm e}^{-2 i x} b}{16 \left (2 i b a +a^{2}-b^{2}\right ) \left (i b +a \right )}-\frac {{\mathrm e}^{-2 i x} a}{16 \left (2 i b a +a^{2}-b^{2}\right ) \left (i b +a \right )}+\frac {{\mathrm e}^{-4 i x}}{128 i b a +64 a^{2}-64 b^{2}}+\frac {6 i a^{4} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {6 i a^{2} b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i a^{3} b^{3}}{\left (i b +a \right )^{3} \left (-i b +a \right )^{4} \left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )}-\frac {3 a^{4} b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {3 a^{2} b^{4} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}\) | \(487\) |
| norman | \(\text {Expression too large to display}\) | \(1658\) |
a^3*b^2/(a^2+b^2)^3/(a+b*tan(x))-3*a^2*b^2*(a^2-b^2)/(a^2+b^2)^4*ln(a+b*ta n(x))+1/(a^2+b^2)^4*(((1/2*a^3*b^3-3/4*a*b^5+5/4*a^5*b)*tan(x)^3+(-1/2*a^6 +a^4*b^2+3/2*a^2*b^4)*tan(x)^2+(3/4*a^5*b-1/2*a^3*b^3-5/4*a*b^5)*tan(x)-1/ 4*a^6+5/4*a^4*b^2+5/4*a^2*b^4-1/4*b^6)/(1+tan(x)^2)^2+3/4*a*b*(1/2*(4*a^3* b-4*a*b^3)*ln(1+tan(x)^2)+(-a^4+6*a^2*b^2-b^4)*arctan(tan(x))))
Time = 0.31 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.77 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {8 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )^{5} - 8 \, {\left (2 \, a^{7} + 3 \, a^{5} b^{2} - a b^{6}\right )} \cos \left (x\right )^{3} + {\left (5 \, a^{7} + 21 \, a^{5} b^{2} + 27 \, a^{3} b^{4} - 21 \, a b^{6} - 24 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} x\right )} \cos \left (x\right ) - 48 \, {\left ({\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (x\right ) + {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) + {\left (5 \, a^{6} b - 51 \, a^{4} b^{3} - 21 \, a^{2} b^{5} + 3 \, b^{7} - 8 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{4} + 24 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (x\right )^{2} - 24 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} x\right )} \sin \left (x\right )}{32 \, {\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (x\right ) + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \sin \left (x\right )\right )}} \]
1/32*(8*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(x)^5 - 8*(2*a^7 + 3*a^5* b^2 - a*b^6)*cos(x)^3 + (5*a^7 + 21*a^5*b^2 + 27*a^3*b^4 - 21*a*b^6 - 24*( a^6*b - 6*a^4*b^3 + a^2*b^5)*x)*cos(x) - 48*((a^5*b^2 - a^3*b^4)*cos(x) + (a^4*b^3 - a^2*b^5)*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + (5*a^6*b - 51*a^4*b^3 - 21*a^2*b^5 + 3*b^7 - 8*(a^6*b + 3*a^4*b^ 3 + 3*a^2*b^5 + b^7)*cos(x)^4 + 24*(a^6*b + 2*a^4*b^3 + a^2*b^5)*cos(x)^2 - 24*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*x)*sin(x))/((a^9 + 4*a^7*b^2 + 6*a^5*b^ 4 + 4*a^3*b^6 + a*b^8)*cos(x) + (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*sin(x))
Timed out. \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (200) = 400\).
Time = 0.31 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.17 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {3 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} x}{4 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac {3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac {a^{5} - 10 \, a^{3} b^{2} + a b^{4} - 3 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (x\right )^{4} - 3 \, {\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (x\right )^{3} + {\left (2 \, a^{5} - 17 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \tan \left (x\right )^{2} - {\left (2 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \tan \left (x\right )}{4 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )^{5} + {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (x\right )^{4} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )^{3} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (x\right )^{2} + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )\right )}} \]
-3/4*(a^5*b - 6*a^3*b^3 + a*b^5)*x/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^ 6 + b^8) - 3*(a^4*b^2 - a^2*b^4)*log(b*tan(x) + a)/(a^8 + 4*a^6*b^2 + 6*a^ 4*b^4 + 4*a^2*b^6 + b^8) + 3/2*(a^4*b^2 - a^2*b^4)*log(tan(x)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 1/4*(a^5 - 10*a^3*b^2 + a*b^4 - 3*(3*a^3*b^2 - a*b^4)*tan(x)^4 - 3*(a^4*b + a^2*b^3)*tan(x)^3 + (2*a^5 - 17*a^3*b^2 + 5*a*b^4)*tan(x)^2 - (2*a^4*b + a^2*b^3 - b^5)*tan(x))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*ta n(x)^5 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*tan(x)^4 + 2*(a^6*b + 3*a^4 *b^3 + 3*a^2*b^5 + b^7)*tan(x)^3 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6) *tan(x)^2 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*tan(x))
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (200) = 400\).
Time = 0.29 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.07 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {3 \, {\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} x}{4 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} + \frac {3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac {3 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {3 \, a^{4} b^{3} \tan \left (x\right ) - 3 \, a^{2} b^{5} \tan \left (x\right ) + 4 \, a^{5} b^{2} - 2 \, a^{3} b^{4}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (x\right ) + a\right )}} - \frac {9 \, a^{4} b^{2} \tan \left (x\right )^{4} - 9 \, a^{2} b^{4} \tan \left (x\right )^{4} - 5 \, a^{5} b \tan \left (x\right )^{3} - 2 \, a^{3} b^{3} \tan \left (x\right )^{3} + 3 \, a b^{5} \tan \left (x\right )^{3} + 2 \, a^{6} \tan \left (x\right )^{2} + 14 \, a^{4} b^{2} \tan \left (x\right )^{2} - 24 \, a^{2} b^{4} \tan \left (x\right )^{2} - 3 \, a^{5} b \tan \left (x\right ) + 2 \, a^{3} b^{3} \tan \left (x\right ) + 5 \, a b^{5} \tan \left (x\right ) + a^{6} + 4 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + b^{6}}{4 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \]
-3/4*(a^5*b - 6*a^3*b^3 + a*b^5)*x/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^ 6 + b^8) + 3/2*(a^4*b^2 - a^2*b^4)*log(tan(x)^2 + 1)/(a^8 + 4*a^6*b^2 + 6* a^4*b^4 + 4*a^2*b^6 + b^8) - 3*(a^4*b^3 - a^2*b^5)*log(abs(b*tan(x) + a))/ (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) + (3*a^4*b^3*tan(x) - 3* a^2*b^5*tan(x) + 4*a^5*b^2 - 2*a^3*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4* a^2*b^6 + b^8)*(b*tan(x) + a)) - 1/4*(9*a^4*b^2*tan(x)^4 - 9*a^2*b^4*tan(x )^4 - 5*a^5*b*tan(x)^3 - 2*a^3*b^3*tan(x)^3 + 3*a*b^5*tan(x)^3 + 2*a^6*tan (x)^2 + 14*a^4*b^2*tan(x)^2 - 24*a^2*b^4*tan(x)^2 - 3*a^5*b*tan(x) + 2*a^3 *b^3*tan(x) + 5*a*b^5*tan(x) + a^6 + 4*a^4*b^2 - 14*a^2*b^4 + b^6)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(tan(x)^2 + 1)^2)
Time = 39.73 (sec) , antiderivative size = 8198, normalized size of antiderivative = 39.04 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Too large to display} \]
((tan(x/2)^4*(a*b^2 + 4*a^3))/(a^4 + b^4 + 2*a^2*b^2) - (tan(x/2)^6*(a*b^2 + 4*a^3))/(a^4 + b^4 + 2*a^2*b^2) - (3*a*b^2*tan(x/2)^2)/(a^4 + b^4 + 2*a ^2*b^2) + (3*a*b^2*tan(x/2)^8)/(a^4 + b^4 + 2*a^2*b^2) + (3*b*tan(x/2)^9*( a^4 - 3*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (3*b*tan(x/2)* (a^4 - 3*a^2*b^2))/(2*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) + (4*b*tan(x/2) ^3*(a^4 + b^4 - 4*a^2*b^2))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) + (4*b*t an(x/2)^7*(a^4 + b^4 - 4*a^2*b^2))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) - (3*b*tan(x/2)^5*(a^4 + 13*a^2*b^2))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) )/(a + 2*b*tan(x/2) + 3*a*tan(x/2)^2 + 2*a*tan(x/2)^4 - 2*a*tan(x/2)^6 - 3 *a*tan(x/2)^8 - a*tan(x/2)^10 + 8*b*tan(x/2)^3 + 12*b*tan(x/2)^5 + 8*b*tan (x/2)^7 + 2*b*tan(x/2)^9) + (log(a + 2*b*tan(x/2) - a*tan(x/2)^2)*(3*a^2*b ^4 - 3*a^4*b^2))/(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) - (log(1/ (cos(x) + 1))*(96*a^2*b^4 - 96*a^4*b^2))/(2*(16*a^8 + 16*b^8 + 64*a^2*b^6 + 96*a^4*b^4 + 64*a^6*b^2)) + (3*a*b*atan((tan(x/2)*((((6*(45*a^7*b^10 - 1 8*a^5*b^12 - 135*a^9*b^8 + 99*a^11*b^6 + 9*a^13*b^4))/(a^18 + b^18 + 9*a^2 *b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12* b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (((6*(6*a^3*b^16 - 153*a^5*b^14 - 180*a^ 7*b^12 + 357*a^9*b^10 + 534*a^11*b^8 + 81*a^13*b^6 - 72*a^15*b^4 + 3*a^17* b^2))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - ((96*a^2*b^...