Integrand size = 20, antiderivative size = 131 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{2 \left (a^2+b^2\right )^3}+\frac {2 a b \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac {\left (-a^2+b^2\right ) \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac {a b \sin ^2(x)}{\left (a^2+b^2\right )^2}+\frac {a b^2 \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \] Output:
1/2*(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^3+2*a*b*(a^2-b^2)*ln(a*cos(x)+b*sin(x) )/(a^2+b^2)^3+1/2*(-a^2+b^2)*cos(x)*sin(x)/(a^2+b^2)^2+a*b*sin(x)^2/(a^2+b ^2)^2+a*b^2*sin(x)/(a^2+b^2)^2/(a*cos(x)+b*sin(x))
Time = 1.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {\sin (x)}{8 a (a \cos (x)+b \sin (x))}-\frac {-4 \left (a^4-6 a^2 b^2+b^4\right ) x+4 a b \left (a^2+b^2\right ) \cos (2 x)-16 a b \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))+\frac {\left (a^2+b^2\right ) \left (a^4-6 a^2 b^2+b^4\right ) \sin (x)}{a (a \cos (x)+b \sin (x))}+2 \left (a^4-b^4\right ) \sin (2 x)}{8 \left (a^2+b^2\right )^3} \] Input:
Integrate[(Cos[x]^2*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]
Output:
Sin[x]/(8*a*(a*Cos[x] + b*Sin[x])) - (-4*(a^4 - 6*a^2*b^2 + b^4)*x + 4*a*b *(a^2 + b^2)*Cos[2*x] - 16*a*b*(a^2 - b^2)*Log[a*Cos[x] + b*Sin[x]] + ((a^ 2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Sin[x])/(a*(a*Cos[x] + b*Sin[x])) + 2*(a^ 4 - b^4)*Sin[2*x])/(8*(a^2 + b^2)^3)
Leaf count is larger than twice the leaf count of optimal. \(336\) vs. \(2(131)=262\).
Time = 2.50 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.56, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.050, Rules used = {3042, 3590, 3042, 3588, 3042, 3044, 15, 3115, 24, 3576, 3042, 3577, 3042, 3590, 3042, 3554, 3576, 3042, 3577, 3042, 3612}
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.
| \(\displaystyle \int \frac {\sin ^2(x) \cos ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^2 \cos (x)^2}{(a \cos (x)+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {b \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {b \left (\frac {b \int \cos ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \int \sin ^2(x)dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (\frac {a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle \frac {b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (-\frac {a b \int \frac {1}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (-\frac {a b \int \frac {1}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3554 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \sin (x)}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \sin (x)}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \sin (x)}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {a \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \sin (x)}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {a \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \sin (x)}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle \frac {a \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (-\frac {b \sin (x)}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {a \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
Input:
Int[(Cos[x]^2*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]
Output:
-((a*b*((a*((b*x)/(a^2 + b^2) - (a*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2))) /(a^2 + b^2) + (b*((a*x)/(a^2 + b^2) + (b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)))/(a^2 + b^2) - (b*Sin[x])/((a^2 + b^2)*(a*Cos[x] + b*Sin[x]))))/(a^ 2 + b^2)) + (a*(-((a*b*((b*x)/(a^2 + b^2) - (a*Log[a*Cos[x] + b*Sin[x]])/( a^2 + b^2)))/(a^2 + b^2)) + (b*Sin[x]^2)/(2*(a^2 + b^2)) + (a*(x/2 - (Cos[ x]*Sin[x])/2))/(a^2 + b^2)))/(a^2 + b^2) + (b*(-((a*b*((a*x)/(a^2 + b^2) + (b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)))/(a^2 + b^2)) + (a*Sin[x]^2)/(2 *(a^2 + b^2)) + (b*(x/2 + (Cos[x]*Sin[x])/2))/(a^2 + b^2)))/(a^2 + b^2)
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[((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[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-2), x _Symbol] :> Simp[Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x])), x] / ; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
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]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x _Symbol] :> Simp[(b*B + c*C)*(x/(b^2 + c^2)), x] + Simp[(c*B - b*C)*(Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C ), 0]
Time = 0.61 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(-\frac {b \,a^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (x \right )\right )}+\frac {2 b a \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-\frac {a^{4}}{2}+\frac {b^{4}}{2}\right ) \tan \left (x \right )-a^{3} b -b^{3} a}{\tan \left (x \right )^{2}+1}+\frac {\left (-4 a^{3} b +4 b^{3} a \right ) \ln \left (\tan \left (x \right )^{2}+1\right )}{4}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (x \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{3}}\) | \(143\) |
| parallelrisch | \(\frac {16 a b \left (a -b \right ) \left (a +b \right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )-16 a b \left (a -b \right ) \left (a +b \right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right )-b \cos \left (3 x \right ) \left (a^{2}+b^{2}\right )^{2}-a \left (a^{2}+b^{2}\right )^{2} \sin \left (3 x \right )+\left (4 a^{5} x -24 a^{3} b^{2} x +4 a \,b^{4} x +a^{4} b +2 a^{2} b^{3}+b^{5}\right ) \cos \left (x \right )-\sin \left (x \right ) \left (-4 a^{4} b x +24 a^{2} b^{3} x -4 b^{5} x +a^{5}-14 a^{3} b^{2}-15 b^{4} a \right )}{8 \left (a^{2}+b^{2}\right )^{3} \left (a \cos \left (x \right )+b \sin \left (x \right )\right )}\) | \(211\) |
| risch | \(-\frac {i x b}{2 \left (3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}\right )}-\frac {x a}{2 \left (3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}\right )}+\frac {i {\mathrm e}^{2 i x}}{-16 i a b +8 a^{2}-8 b^{2}}-\frac {i {\mathrm e}^{-2 i x}}{8 \left (2 i a b +a^{2}-b^{2}\right )}-\frac {4 i a^{3} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {4 i a \,b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i a^{2} b^{2}}{\left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right ) \left (i b +a \right )^{2} \left (-i b +a \right )^{3}}+\frac {2 a^{3} b \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(340\) |
| norman | \(\frac {\frac {2 b \tan \left (\frac {x}{2}\right )^{8}}{a^{2}+b^{2}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \tan \left (\frac {x}{2}\right )^{6}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 b \tan \left (\frac {x}{2}\right )^{4}}{a^{2}+b^{2}}+\frac {2 b \tan \left (\frac {x}{2}\right )^{6}}{a^{2}+b^{2}}-\frac {2 b \tan \left (\frac {x}{2}\right )^{2}}{a^{2}+b^{2}}-\frac {\left (-a^{4}+3 a^{2} b^{2}\right ) \tan \left (\frac {x}{2}\right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (-a^{4}+3 a^{2} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{9}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 \left (a^{4}+13 a^{2} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{5}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {16 b^{2} a \tan \left (\frac {x}{2}\right )^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {16 b^{2} a \tan \left (\frac {x}{2}\right )^{7}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {x}{2}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {4 b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{3}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {4 b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{7}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{9}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \tan \left (\frac {x}{2}\right )^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \tan \left (\frac {x}{2}\right )^{8}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \tan \left (\frac {x}{2}\right )^{10}}{2 a^{6}+6 a^{4} b^{2}+6 a^{2} b^{4}+2 b^{6}}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(933\) |
Input:
int(cos(x)^2*sin(x)^2/(a*cos(x)+b*sin(x))^2,x,method=_RETURNVERBOSE)
Output:
-b*a^2/(a^2+b^2)^2/(a+b*tan(x))+2*b*a*(a^2-b^2)/(a^2+b^2)^3*ln(a+b*tan(x)) +1/(a^2+b^2)^3*(((-1/2*a^4+1/2*b^4)*tan(x)-a^3*b-b^3*a)/(tan(x)^2+1)+1/4*( -4*a^3*b+4*a*b^3)*ln(tan(x)^2+1)+1/2*(a^4-6*a^2*b^2+b^4)*arctan(tan(x)))
Time = 0.09 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{3} + {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} x\right )} \cos \left (x\right ) - 2 \, {\left ({\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - {\left (3 \, a^{3} b^{2} + a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2} + {\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} x\right )} \sin \left (x\right )}{2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(cos(x)^2*sin(x)^2/(a*cos(x)+b*sin(x))^2,x, algorithm="fricas")
Output:
-1/2*((a^4*b + 2*a^2*b^3 + b^5)*cos(x)^3 + (a^2*b^3 - b^5 - (a^5 - 6*a^3*b ^2 + a*b^4)*x)*cos(x) - 2*((a^4*b - a^2*b^3)*cos(x) + (a^3*b^2 - a*b^4)*si n(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - (3*a^3*b^2 + a*b^4 - (a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^2 + (a^4*b - 6*a^2*b^3 + b^5)*x) *sin(x))/((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(x) + (a^6*b + 3*a^4*b^ 3 + 3*a^2*b^5 + b^7)*sin(x))
Timed out. \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \] Input:
integrate(cos(x)**2*sin(x)**2/(a*cos(x)+b*sin(x))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (127) = 254\).
Time = 0.11 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.96 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} x}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, a^{2} b + {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (x\right )^{2} + {\left (a^{3} + a b^{2}\right )} \tan \left (x\right )}{2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \left (x\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )\right )}} \] Input:
integrate(cos(x)^2*sin(x)^2/(a*cos(x)+b*sin(x))^2,x, algorithm="maxima")
Output:
1/2*(a^4 - 6*a^2*b^2 + b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(a^3 *b - a*b^3)*log(b*tan(x) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^3*b - a*b^3)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/2*(4*a ^2*b + (3*a^2*b - b^3)*tan(x)^2 + (a^3 + a*b^2)*tan(x))/(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*tan(x)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*ta n(x)^2 + (a^4*b + 2*a^2*b^3 + b^5)*tan(x))
Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.67 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} x}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {3 \, a^{2} b \tan \left (x\right )^{2} - b^{3} \tan \left (x\right )^{2} + a^{3} \tan \left (x\right ) + a b^{2} \tan \left (x\right ) + 4 \, a^{2} b}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (x\right )^{3} + a \tan \left (x\right )^{2} + b \tan \left (x\right ) + a\right )}} \] Input:
integrate(cos(x)^2*sin(x)^2/(a*cos(x)+b*sin(x))^2,x, algorithm="giac")
Output:
1/2*(a^4 - 6*a^2*b^2 + b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^3*b - a*b^3)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(a^3*b ^2 - a*b^4)*log(abs(b*tan(x) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 1/2*(3*a^2*b*tan(x)^2 - b^3*tan(x)^2 + a^3*tan(x) + a*b^2*tan(x) + 4*a^2* b)/((a^4 + 2*a^2*b^2 + b^4)*(b*tan(x)^3 + a*tan(x)^2 + b*tan(x) + a))
Time = 27.55 (sec) , antiderivative size = 6012, normalized size of antiderivative = 45.89 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Too large to display} \] Input:
int((cos(x)^2*sin(x)^2)/(a*cos(x) + b*sin(x))^2,x)
Output:
((tan(x/2)^5*(3*a*b^2 - a^3))/(a^4 + b^4 + 2*a^2*b^2) + (2*b*tan(x/2)^2)/( a^2 + b^2) - (2*b*tan(x/2)^4)/(a^2 + b^2) + (tan(x/2)*(3*a*b^2 - a^3))/(a^ 2 + b^2)^2 + (2*tan(x/2)^3*(5*a*b^2 + a^3))/(a^2 + b^2)^2)/(a + 2*b*tan(x/ 2) + a*tan(x/2)^2 - a*tan(x/2)^4 - a*tan(x/2)^6 + 4*b*tan(x/2)^3 + 2*b*tan (x/2)^5) - (log(a + 2*b*tan(x/2) - a*tan(x/2)^2)*(2*a*b^3 - 2*a^3*b))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (log(1/(cos(x) + 1))*(16*a*b^3 - 16*a^3* b))/(2*(4*a^6 + 4*b^6 + 12*a^2*b^4 + 12*a^4*b^2)) - (atan((tan(x/2)*(((((( ((8*(4*a^2*b^13 - 20*a^14*b + 48*a^4*b^11 + 132*a^6*b^9 + 128*a^8*b^7 + 12 *a^10*b^5 - 48*a^12*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6* b^6 + 15*a^8*b^4 + 6*a^10*b^2) - (4*(16*a*b^3 - 16*a^3*b)*(12*a*b^16 + 84* a^3*b^14 + 252*a^5*b^12 + 420*a^7*b^10 + 420*a^9*b^8 + 252*a^11*b^6 + 84*a ^13*b^4 + 12*a^15*b^2))/((4*a^6 + 4*b^6 + 12*a^2*b^4 + 12*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))* (2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(2*(a^2 + b^2)*(a^4 + b^4 + 2*a^2 *b^2)) - (2*(16*a*b^3 - 16*a^3*b)*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2)* (12*a*b^16 + 84*a^3*b^14 + 252*a^5*b^12 + 420*a^7*b^10 + 420*a^9*b^8 + 252 *a^11*b^6 + 84*a^13*b^4 + 12*a^15*b^2))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^ 2)*(4*a^6 + 4*b^6 + 12*a^2*b^4 + 12*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 1 5*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(16*a*b^3 - 16*a^3*b)) /(2*(4*a^6 + 4*b^6 + 12*a^2*b^4 + 12*a^4*b^2)) - (((8*(2*a*b^12 + a^13 ...
Time = 0.17 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.17 \[ \int \frac {\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {-4 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) a^{4} b^{2}+4 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) a^{2} b^{4}+4 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -2 \tan \left (\frac {x}{2}\right ) b -a \right ) a^{4} b^{2}-4 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -2 \tan \left (\frac {x}{2}\right ) b -a \right ) a^{2} b^{4}+\cos \left (x \right ) \sin \left (x \right )^{2} a^{4} b^{2}+2 \cos \left (x \right ) \sin \left (x \right )^{2} a^{2} b^{4}+\cos \left (x \right ) \sin \left (x \right )^{2} b^{6}+\cos \left (x \right ) a^{6}+\cos \left (x \right ) a^{5} b x -2 \cos \left (x \right ) a^{4} b^{2}-6 \cos \left (x \right ) a^{3} b^{3} x -3 \cos \left (x \right ) a^{2} b^{4}+\cos \left (x \right ) a \,b^{5} x -4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) \sin \left (x \right ) a^{3} b^{3}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) \sin \left (x \right ) a \,b^{5}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -2 \tan \left (\frac {x}{2}\right ) b -a \right ) \sin \left (x \right ) a^{3} b^{3}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -2 \tan \left (\frac {x}{2}\right ) b -a \right ) \sin \left (x \right ) a \,b^{5}+\sin \left (x \right )^{3} a^{5} b +2 \sin \left (x \right )^{3} a^{3} b^{3}+\sin \left (x \right )^{3} a \,b^{5}+\sin \left (x \right ) a^{4} b^{2} x -6 \sin \left (x \right ) a^{2} b^{4} x +\sin \left (x \right ) b^{6} x}{2 b \left (\cos \left (x \right ) a^{7}+3 \cos \left (x \right ) a^{5} b^{2}+3 \cos \left (x \right ) a^{3} b^{4}+\cos \left (x \right ) a \,b^{6}+\sin \left (x \right ) a^{6} b +3 \sin \left (x \right ) a^{4} b^{3}+3 \sin \left (x \right ) a^{2} b^{5}+\sin \left (x \right ) b^{7}\right )} \] Input:
int(cos(x)^2*sin(x)^2/(a*cos(x)+b*sin(x))^2,x)
Output:
( - 4*cos(x)*log(tan(x/2)**2 + 1)*a**4*b**2 + 4*cos(x)*log(tan(x/2)**2 + 1 )*a**2*b**4 + 4*cos(x)*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*a**4*b**2 - 4 *cos(x)*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*a**2*b**4 + cos(x)*sin(x)**2 *a**4*b**2 + 2*cos(x)*sin(x)**2*a**2*b**4 + cos(x)*sin(x)**2*b**6 + cos(x) *a**6 + cos(x)*a**5*b*x - 2*cos(x)*a**4*b**2 - 6*cos(x)*a**3*b**3*x - 3*co s(x)*a**2*b**4 + cos(x)*a*b**5*x - 4*log(tan(x/2)**2 + 1)*sin(x)*a**3*b**3 + 4*log(tan(x/2)**2 + 1)*sin(x)*a*b**5 + 4*log(tan(x/2)**2*a - 2*tan(x/2) *b - a)*sin(x)*a**3*b**3 - 4*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)* a*b**5 + sin(x)**3*a**5*b + 2*sin(x)**3*a**3*b**3 + sin(x)**3*a*b**5 + sin (x)*a**4*b**2*x - 6*sin(x)*a**2*b**4*x + sin(x)*b**6*x)/(2*b*(cos(x)*a**7 + 3*cos(x)*a**5*b**2 + 3*cos(x)*a**3*b**4 + cos(x)*a*b**6 + sin(x)*a**6*b + 3*sin(x)*a**4*b**3 + 3*sin(x)*a**2*b**5 + sin(x)*b**7))