Integrand size = 18, antiderivative size = 128 \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {a b \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b^2 \left (3 a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac {a b \cos (x) \sin (x)}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac {a b^2 \cos (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]
-a*b*(a^2-3*b^2)*x/(a^2+b^2)^3-b^2*(3*a^2-b^2)*ln(a*cos(x)+b*sin(x))/(a^2+ b^2)^3+a*b*cos(x)*sin(x)/(a^2+b^2)^2+1/2*(a^2-b^2)*sin(x)^2/(a^2+b^2)^2+a* b^2*cos(x)/(a^2+b^2)^2/(a*cos(x)+b*sin(x))
Result contains complex when optimal does not.
Time = 1.63 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {-4 i b^2 \left (-3 a^2+b^2\right ) \arctan (\tan (x)) (a \cos (x)+b \sin (x))-a \cos (x) \left (\left (a^4-b^4\right ) \cos (2 x)+2 b \left (2 (a+i b)^3 x-b \left (-3 a^2+b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )-a \left (a^2+b^2\right ) \sin (2 x)\right )\right )+b \sin (x) \left (\left (-a^4+b^4\right ) \cos (2 x)+2 b \left (-2 (a+i b) \left (a^2 x-b^2 (i+x)+a (b+2 i b x)\right )+\left (-3 a^2 b+b^3\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+a \left (a^2+b^2\right ) \sin (2 x)\right )\right )}{4 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
((-4*I)*b^2*(-3*a^2 + b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]) - a*Cos[x] *((a^4 - b^4)*Cos[2*x] + 2*b*(2*(a + I*b)^3*x - b*(-3*a^2 + b^2)*Log[(a*Co s[x] + b*Sin[x])^2] - a*(a^2 + b^2)*Sin[2*x])) + b*Sin[x]*((-a^4 + b^4)*Co s[2*x] + 2*b*(-2*(a + I*b)*(a^2*x - b^2*(I + x) + a*(b + (2*I)*b*x)) + (-3 *a^2*b + b^3)*Log[(a*Cos[x] + b*Sin[x])^2] + a*(a^2 + b^2)*Sin[2*x])))/(4* (a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))
Leaf count is larger than twice the leaf count of optimal. \(293\) vs. \(2(128)=256\).
Time = 2.19 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.29, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.389, Rules used = {3042, 3590, 3042, 3565, 3042, 3579, 3042, 3115, 24, 3577, 3042, 3588, 3042, 3044, 15, 3115, 24, 3577, 3042, 3612, 3964, 3042, 4014, 3042, 4013}
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.
| \(\displaystyle \int \frac {\sin (x) \cos ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x) \cos (x)^3}{(a \cos (x)+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {b \int \frac {\cos ^3(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3565 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {a \int \cos ^2(x)dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {a \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \cos ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle -\frac {a b \int \frac {1}{(a+b \tan (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b^2 \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle -\frac {a b \left (\frac {\int \frac {a-b \tan (x)}{a+b \tan (x)}dx}{a^2+b^2}-\frac {b}{\left (a^2+b^2\right ) (a+b \tan (x))}\right )}{a^2+b^2}+\frac {b \left (\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b^2 \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {\int \frac {a-b \tan (x)}{a+b \tan (x)}dx}{a^2+b^2}-\frac {b}{\left (a^2+b^2\right ) (a+b \tan (x))}\right )}{a^2+b^2}+\frac {b \left (\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b^2 \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle -\frac {a b \left (\frac {\frac {2 a b \int \frac {b-a \tan (x)}{a+b \tan (x)}dx}{a^2+b^2}+\frac {x \left (a^2-b^2\right )}{a^2+b^2}}{a^2+b^2}-\frac {b}{\left (a^2+b^2\right ) (a+b \tan (x))}\right )}{a^2+b^2}+\frac {b \left (\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b^2 \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {\frac {2 a b \int \frac {b-a \tan (x)}{a+b \tan (x)}dx}{a^2+b^2}+\frac {x \left (a^2-b^2\right )}{a^2+b^2}}{a^2+b^2}-\frac {b}{\left (a^2+b^2\right ) (a+b \tan (x))}\right )}{a^2+b^2}+\frac {b \left (\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b^2 \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b \left (\frac {b \cos ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {b^2 \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {\frac {x \left (a^2-b^2\right )}{a^2+b^2}+\frac {2 a b \log (a \cos (x)+b \sin (x))}{a^2+b^2}}{a^2+b^2}-\frac {b}{\left (a^2+b^2\right ) (a+b \tan (x))}\right )}{a^2+b^2}\) |
(b*((b*Cos[x]^2)/(2*(a^2 + b^2)) + (b^2*((a*x)/(a^2 + b^2) + (b*Log[a*Cos[ x] + b*Sin[x]])/(a^2 + b^2)))/(a^2 + b^2) + (a*(x/2 + (Cos[x]*Sin[x])/2))/ (a^2 + b^2)))/(a^2 + b^2) + (a*(-((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) - (a*b*((((a^2 - b^2)*x)/(a^2 + b^2) + (2*a*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2))/(a^2 + b^2) - b/((a^2 + b^2)*(a + b*Tan[x]))))/(a^2 + b^2)
3.3.90.3.1 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[((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_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && 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_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b*(Cos[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a/(a^2 + b^2) Int[Cos[c + d*x]^(m - 1), x], x] + Simp[b^2/(a^2 + b^2) Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[ c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
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]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.86 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(\frac {a \,b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (x \right )\right )}-\frac {b^{2} \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3} b +a \,b^{3}\right ) \tan \left (x \right )-\frac {a^{4}}{2}+\frac {b^{4}}{2}}{1+\tan \left (x \right )^{2}}+b \left (\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan \left (x \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{\left (a^{2}+b^{2}\right )^{3}}\) | \(141\) |
| parallelrisch | \(\frac {-24 b^{2} \left (a^{2}-\frac {b^{2}}{3}\right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+24 b^{2} \left (a^{2}-\frac {b^{2}}{3}\right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-a \left (a^{2}+b^{2}\right )^{2} \cos \left (3 x \right )+b \left (a^{2}+b^{2}\right )^{2} \sin \left (3 x \right )+a \left (-8 a^{3} b x +24 a \,b^{3} x +a^{4}+2 a^{2} b^{2}+b^{4}\right ) \cos \left (x \right )+5 b \sin \left (x \right ) \left (-\frac {8}{5} a^{3} b x +\frac {24}{5} a \,b^{3} x +a^{4}-\frac {6}{5} a^{2} b^{2}-\frac {11}{5} b^{4}\right )}{8 \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \left (a^{2}+b^{2}\right )^{3}}\) | \(196\) |
| risch | \(-\frac {i x b}{i a^{3}-3 i a \,b^{2}+3 a^{2} b -b^{3}}-\frac {{\mathrm e}^{2 i x}}{8 \left (-2 i b a +a^{2}-b^{2}\right )}-\frac {{\mathrm e}^{-2 i x}}{8 \left (2 i b a +a^{2}-b^{2}\right )}+\frac {6 i a^{2} x \,b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i b^{4} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i a \,b^{3}}{\left (-i a +b \right )^{2} \left (i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i x}+i a \,{\mathrm e}^{2 i x}-b +i a \right )}-\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right ) b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {b^{4} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(308\) |
| norman | \(\frac {\frac {2 a \tan \left (\frac {x}{2}\right )^{8}}{a^{2}+b^{2}}+\frac {b \,a^{2} \left (a^{2}-3 b^{2}\right ) x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 a \tan \left (\frac {x}{2}\right )^{4}}{a^{2}+b^{2}}+\frac {2 a \tan \left (\frac {x}{2}\right )^{6}}{a^{2}+b^{2}}-\frac {2 a \tan \left (\frac {x}{2}\right )^{2}}{a^{2}+b^{2}}-\frac {2 b \left (a^{3}-a \,b^{2}\right ) \tan \left (\frac {x}{2}\right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \left (a^{3}-a \,b^{2}\right ) \tan \left (\frac {x}{2}\right )^{9}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 \left (a^{3}-3 a \,b^{2}\right ) b \tan \left (\frac {x}{2}\right )^{3}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 \left (a^{3}-3 a \,b^{2}\right ) b \tan \left (\frac {x}{2}\right )^{7}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 \left (a^{3}-5 a \,b^{2}\right ) b \tan \left (\frac {x}{2}\right )^{5}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b \,a^{2} \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 b \,a^{2} \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 b \,a^{2} \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 b \,a^{2} \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{8}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {b \,a^{2} \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{10}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 b^{2} a \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {8 b^{2} a \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{3}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {12 b^{2} a \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {8 b^{2} a \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{7}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 b^{2} a \left (a^{2}-3 b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{9}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+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 {b^{2} \left (3 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 {b^{2} \left (3 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}}\) | \(918\) |
a*b^2/(a^2+b^2)^2/(a+b*tan(x))-b^2*(3*a^2-b^2)/(a^2+b^2)^3*ln(a+b*tan(x))+ 1/(a^2+b^2)^3*(((a^3*b+a*b^3)*tan(x)-1/2*a^4+1/2*b^4)/(1+tan(x)^2)+b*(1/2* (3*a^2*b-b^3)*ln(1+tan(x)^2)+(-a^3+3*a*b^2)*arctan(tan(x))))
Time = 0.28 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.97 \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - {\left (a^{5} + 4 \, a^{3} b^{2} + 7 \, a b^{4} - 4 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} x\right )} \cos \left (x\right ) + 2 \, {\left ({\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right ) + {\left (3 \, a^{2} b^{3} - b^{5}\right )} \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - {\left (a^{4} b - 4 \, a^{2} b^{3} - b^{5} + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{2} - 4 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} x\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\right )}} \]
-1/4*(2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^3 - (a^5 + 4*a^3*b^2 + 7*a*b^4 - 4*(a^4*b - 3*a^2*b^3)*x)*cos(x) + 2*((3*a^3*b^2 - a*b^4)*cos(x) + (3*a^2*b ^3 - b^5)*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - (a^4*b - 4*a^2*b^3 - b^5 + 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)^2 - 4*(a^3*b ^2 - 3*a*b^4)*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. \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (126) = 252\).
Time = 0.33 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.00 \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {{\left (a^{3} b - 3 \, a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {4 \, a b^{2} \tan \left (x\right )^{2} - a^{3} + 3 \, a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (x\right )}{2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \left (x\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )\right )}} \]
-(a^3*b - 3*a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b^2 - b^ 4)*log(b*tan(x) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(3*a^2*b^2 - b^4)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(4*a*b^ 2*tan(x)^2 - a^3 + 3*a*b^2 + (a^2*b + b^3)*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)*tan(x)^ 2 + (a^4*b + 2*a^2*b^3 + b^5)*tan(x))
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.67 \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {{\left (a^{3} b - 3 \, a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (3 \, a^{2} b^{3} - b^{5}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac {4 \, a b^{2} \tan \left (x\right )^{2} + a^{2} b \tan \left (x\right ) + b^{3} \tan \left (x\right ) - a^{3} + 3 \, a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (x\right )^{3} + a \tan \left (x\right )^{2} + b \tan \left (x\right ) + a\right )}} \]
-(a^3*b - 3*a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(3*a^2*b^2 - b^4)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b^3 - b^5)*log(abs(b*tan(x) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) + 1/2* (4*a*b^2*tan(x)^2 + a^2*b*tan(x) + b^3*tan(x) - a^3 + 3*a*b^2)/((a^4 + 2*a ^2*b^2 + b^4)*(b*tan(x)^3 + a*tan(x)^2 + b*tan(x) + a))
Time = 30.83 (sec) , antiderivative size = 5428, normalized size of antiderivative = 42.41 \[ \int \frac {\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Too large to display} \]
((2*a*tan(x/2)^2)/(a^2 + b^2) - (8*b^3*tan(x/2)^3)/(a^2 + b^2)^2 - (2*a*ta n(x/2)^4)/(a^2 + b^2) + (2*b*tan(x/2)*(a^2 - b^2))/(a^2 + b^2)^2 + (2*b*ta n(x/2)^5*(a^2 - b^2))/(a^4 + b^4 + 2*a^2*b^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)*(b^4 - 3*a^2*b^2))/(a^6 + b^6 + 3*a^2 *b^4 + 3*a^4*b^2) - (log(1/(cos(x) + 1))*(2*b^4 - 6*a^2*b^2))/(2*(a^6 + b^ 6 + 3*a^2*b^4 + 3*a^4*b^2)) - (2*a*b*atan((tan(x/2)*((((((a*b*((32*(a*b^14 + 9*a^3*b^12 + 18*a^5*b^10 + 2*a^7*b^8 - 27*a^9*b^6 - 27*a^11*b^4 - 8*a^1 3*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) - (16*(2*b^4 - 6*a^2*b^2)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^ 12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2)) /((a^6 + b^6 + 3*a^2*b^4 + 3*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)))*(a^2 - 3*b^2))/(a^6 + b^6 + 3 *a^2*b^4 + 3*a^4*b^2) - (16*a*b*(a^2 - 3*b^2)*(2*b^4 - 6*a^2*b^2)*(3*a*b^1 6 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^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*b^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*b*(a^2 - 3*b^2)*((32*(3*a*b^12 - 21*a^3*b^10 - 34*a^5*b^8 + 6*a^7*b^6 + 15*a^9*b^4 - a^11*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*...