Integrand size = 18, antiderivative size = 129 \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {b \left (3 a^3-a b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a^2 \left (a^2-3 b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac {a b \cos (x) \sin (x)}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {a^2 b \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]
b*(3*a^3-a*b^2)*x/(a^2+b^2)^3-a^2*(a^2-3*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^2 *b*sin(x)/(a^2+b^2)^2/(a*cos(x)+b*sin(x))
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.75 \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {4 i a^2 \left (a^2-3 b^2\right ) \arctan (\tan (x)) (a \cos (x)+b \sin (x))+a \cos (x) \left (\left (a^4-b^4\right ) \cos (2 x)+2 a \left (2 (i a-b)^3 x-a \left (a^2-3 b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )-b \left (a^2+b^2\right ) \sin (2 x)\right )\right )-b \sin (x) \left (\left (-a^4+b^4\right ) \cos (2 x)+2 a \left (2 \left (a^3 (1+i x)+a b^2 (1-3 i x)-3 a^2 b x+b^3 x\right )+a \left (a^2-3 b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+b \left (a^2+b^2\right ) \sin (2 x)\right )\right )}{4 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
((4*I)*a^2*(a^2 - 3*b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]) + a*Cos[x]*( (a^4 - b^4)*Cos[2*x] + 2*a*(2*(I*a - b)^3*x - a*(a^2 - 3*b^2)*Log[(a*Cos[x ] + b*Sin[x])^2] - b*(a^2 + b^2)*Sin[2*x])) - b*Sin[x]*((-a^4 + b^4)*Cos[2 *x] + 2*a*(2*(a^3*(1 + I*x) + a*b^2*(1 - (3*I)*x) - 3*a^2*b*x + b^3*x) + a *(a^2 - 3*b^2)*Log[(a*Cos[x] + b*Sin[x])^2] + b*(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. \(295\) vs. \(2(129)=258\).
Time = 2.24 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.29, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.444, Rules used = {3042, 3590, 3042, 3564, 3042, 3578, 3042, 3115, 24, 3576, 3042, 3588, 3042, 3044, 15, 3115, 24, 3576, 3042, 3612, 3964, 3042, 4014, 25, 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.
| \(\displaystyle \int \frac {\sin ^3(x) \cos (x)}{(a \cos (x)+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^3 \cos (x)}{(a \cos (x)+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {a \int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\sin ^2(x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\sin (x)^2}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\sin (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3564 |
| \(\displaystyle -\frac {a b \int \frac {1}{(b+a \cot (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\sin (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\sin (x)^3}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3578 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {b \int \sin ^2(x)dx}{a^2+b^2}+\frac {a^2 \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {b \int \sin (x)^2dx}{a^2+b^2}+\frac {a^2 \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \int \frac {\cos (x) \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {a \int \sin ^2(x)dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {b \left (\frac {a \int \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {b \left (\frac {a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {a \left (\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle -\frac {a b \int \frac {1}{\left (b-a \tan \left (x+\frac {\pi }{2}\right )\right )^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle -\frac {a b \left (\frac {\int \frac {b-a \cot (x)}{b+a \cot (x)}dx}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a \cot (x)+b)}\right )}{a^2+b^2}+\frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {\int \frac {b+a \tan \left (x+\frac {\pi }{2}\right )}{b-a \tan \left (x+\frac {\pi }{2}\right )}dx}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a \cot (x)+b)}\right )}{a^2+b^2}+\frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle -\frac {a b \left (\frac {-\frac {2 a b \int -\frac {a-b \cot (x)}{b+a \cot (x)}dx}{a^2+b^2}-\frac {x \left (a^2-b^2\right )}{a^2+b^2}}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a \cot (x)+b)}\right )}{a^2+b^2}+\frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a b \left (\frac {\frac {2 a b \int \frac {a-b \cot (x)}{b+a \cot (x)}dx}{a^2+b^2}-\frac {x \left (a^2-b^2\right )}{a^2+b^2}}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a \cot (x)+b)}\right )}{a^2+b^2}+\frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {\frac {2 a b \int \frac {a+b \tan \left (x+\frac {\pi }{2}\right )}{b-a \tan \left (x+\frac {\pi }{2}\right )}dx}{a^2+b^2}-\frac {x \left (a^2-b^2\right )}{a^2+b^2}}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a \cot (x)+b)}\right )}{a^2+b^2}+\frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b \left (\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}+\frac {a^2 \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \left (\frac {a}{\left (a^2+b^2\right ) (a \cot (x)+b)}+\frac {-\frac {x \left (a^2-b^2\right )}{a^2+b^2}-\frac {2 a b \log (a \cos (x)+b \sin (x))}{a^2+b^2}}{a^2+b^2}\right )}{a^2+b^2}\) |
-((a*b*(a/((a^2 + b^2)*(b + a*Cot[x])) + (-(((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)))/(a^2 + b^2)) + (b*(-((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) + (a*((a^2*((b*x)/(a^2 + b^2) - (a*Log[a*Co s[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)
3.3.86.3.1 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[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[(b + a*Cot[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && 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[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a^2/(a^2 + b^2) Int[Sin[c + d*x]^(m - 2)/(a *Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Simp[b/(a^2 + b^2) Int[Sin[c + d*x]^(m - 1), 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 = 0.82 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {\frac {\left (-a^{3} b -a \,b^{3}\right ) \tan \left (x \right )+\frac {a^{4}}{2}-\frac {b^{4}}{2}}{1+\tan \left (x \right )^{2}}+a \left (\frac {\left (a^{3}-3 a \,b^{2}\right ) \ln \left (1+\tan \left (x \right )^{2}\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{3}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (x \right )\right )}-\frac {a^{2} \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}\) | \(138\) |
| parallelrisch | \(\frac {-8 a^{2} \left (a^{2}-3 b^{2}\right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+8 a^{2} \left (a^{2}-3 b^{2}\right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+a \left (a^{2}+b^{2}\right )^{2} \cos \left (3 x \right )-b \left (a^{2}+b^{2}\right )^{2} \sin \left (3 x \right )-a \left (-24 a^{3} b x +8 a \,b^{3} x +a^{4}+2 a^{2} b^{2}+b^{4}\right ) \cos \left (x \right )-13 b \left (-\frac {24}{13} a^{3} b x +\frac {8}{13} a \,b^{3} x +a^{4}+\frac {10}{13} a^{2} b^{2}-\frac {3}{13} b^{4}\right ) \sin \left (x \right )}{8 \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \left (a^{2}+b^{2}\right )^{3}}\) | \(197\) |
| risch | \(\frac {i a x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}}+\frac {{\mathrm e}^{2 i x}}{-16 i b a +8 a^{2}-8 b^{2}}+\frac {{\mathrm e}^{-2 i x}}{16 i b a +8 a^{2}-8 b^{2}}+\frac {2 i a^{4} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i a^{2} x \,b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i a^{3} b}{\left (i b +a \right )^{2} \left (-i b +a \right )^{3} \left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right ) b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(307\) |
| norman | \(\frac {-\frac {2 a \tan \left (\frac {x}{2}\right )^{8}}{a^{2}+b^{2}}+\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x \tan \left (\frac {x}{2}\right )^{10}}{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 {4 b \,a^{2} \tan \left (\frac {x}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {4 b \,a^{2} \tan \left (\frac {x}{2}\right )^{9}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 b \left (-3 a^{3}+a \,b^{2}\right ) \tan \left (\frac {x}{2}\right )^{3}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 b \left (-3 a^{3}+a \,b^{2}\right ) \tan \left (\frac {x}{2}\right )^{7}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {4 \left (-4 a^{3}+2 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 {2 b^{2} \left (3 a^{2}-b^{2}\right ) a 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} \left (3 a^{2}-b^{2}\right ) a 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} \left (3 a^{2}-b^{2}\right ) a 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} \left (3 a^{2}-b^{2}\right ) a 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} \left (3 a^{2}-b^{2}\right ) a 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 (3 a^{2}-b^{2}\right ) a^{2} b x \tan \left (\frac {x}{2}\right )^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 \left (3 a^{2}-b^{2}\right ) a^{2} b x \tan \left (\frac {x}{2}\right )^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 \left (3 a^{2}-b^{2}\right ) a^{2} b x \tan \left (\frac {x}{2}\right )^{6}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \left (3 a^{2}-b^{2}\right ) a^{2} b x \tan \left (\frac {x}{2}\right )^{8}}{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 {a^{2} \left (a^{2}-3 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 {a^{2} \left (a^{2}-3 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}}\) | \(920\) |
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)+a*(1/2 *(a^3-3*a*b^2)*ln(1+tan(x)^2)+(3*a^2*b-b^3)*arctan(tan(x))))+a^3/(a^2+b^2) ^2/(a+b*tan(x))-a^2*(a^2-3*b^2)/(a^2+b^2)^3*ln(a+b*tan(x))
Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.83 \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - {\left (a^{5} + 3 \, a b^{4} - 4 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} x\right )} \cos \left (x\right ) - 2 \, {\left ({\left (a^{5} - 3 \, a^{3} b^{2}\right )} \cos \left (x\right ) + {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - {\left (5 \, a^{4} b - b^{5} + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{2} - 4 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} x\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\right )}} \]
1/4*(2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^3 - (a^5 + 3*a*b^4 - 4*(3*a^4*b - a^2*b^3)*x)*cos(x) - 2*((a^5 - 3*a^3*b^2)*cos(x) + (a^4*b - 3*a^2*b^3)*sin (x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - (5*a^4*b - b^ 5 + 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)^2 - 4*(3*a^3*b^2 - 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. \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (127) = 254\).
Time = 0.30 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.01 \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\left (3 \, a^{3} b - a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {3 \, a^{3} - a b^{2} + 2 \, {\left (a^{3} - a b^{2}\right )} \tan \left (x\right )^{2} - {\left (a^{2} b + b^{3}\right )} \tan \left (x\right )}{2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \left (x\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )\right )}} \]
(3*a^3*b - a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^4 - 3*a^2*b^2 )*log(b*tan(x) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(a^4 - 3*a^2 *b^2)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(3*a^3 - a*b^2 + 2*(a^3 - a*b^2)*tan(x)^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 = 223, normalized size of antiderivative = 1.73 \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\left (3 \, a^{3} b - a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac {2 \, a^{3} \tan \left (x\right )^{2} - 2 \, a b^{2} \tan \left (x\right )^{2} - a^{2} b \tan \left (x\right ) - b^{3} \tan \left (x\right ) + 3 \, a^{3} - a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (x\right )^{3} + a \tan \left (x\right )^{2} + b \tan \left (x\right ) + a\right )}} \]
(3*a^3*b - a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(a^4 - 3*a^2 *b^2)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^4*b - 3*a ^2*b^3)*log(abs(b*tan(x) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) + 1/2 *(2*a^3*tan(x)^2 - 2*a*b^2*tan(x)^2 - a^2*b*tan(x) - b^3*tan(x) + 3*a^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 = 30.77 (sec) , antiderivative size = 5431, normalized size of antiderivative = 42.10 \[ \int \frac {\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Too large to display} \]
(log(1/(cos(x) + 1))*(2*a^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^ 4*b^2)) - (log(a + 2*b*tan(x/2) - a*tan(x/2)^2)*(a^4 - 3*a^2*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - ((2*a*tan(x/2)^2)/(a^2 + b^2) - (2*a*tan(x/ 2)^4)/(a^2 + b^2) + (4*a^2*b*tan(x/2))/(a^2 + b^2)^2 + (4*a^2*b*tan(x/2)^5 )/(a^4 + b^4 + 2*a^2*b^2) + (4*b*tan(x/2)^3*(a^2 - b^2))/(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) + (2*a*b*atan((((((a*b*((32*(3*a^4*b^11 - a^2*b^13 - 4*a^14*b + 18*a^6*b^9 + 22*a^8*b^7 + 3*a^10*b^5 - 9*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) - (16 *(2*a^4 - 6*a^2*b^2)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((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)))*(3*a^2 - b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b ^2) - (16*a*b*(2*a^4 - 6*a^2*b^2)*(3*a^2 - b^2)*(3*a^16*b + 3*a^2*b^15 + 2 1*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a ^14*b^3))/((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*a^4 - 6*a^2*b^2 ))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*b*((32*(5*a^4*b^9 - 3*a^12 *b + 12*a^6*b^7 + 6*a^8*b^5 - 4*a^10*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) + (((32*(3*a^4*b^11 - a...