Integrand size = 20, antiderivative size = 176 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {a b^2 \left (3 a^2-2 b^2\right ) \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {2 a b \left (a^2-b^2\right ) \cos (x)}{\left (a^2+b^2\right )^3}-\frac {2 a b \cos ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {b^2 \left (3 a^2-b^2\right ) \sin (x)}{\left (a^2+b^2\right )^3}+\frac {\left (a^2-b^2\right ) \sin ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^2 b^3}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
-a*b^2*(3*a^2-2*b^2)*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))/(a^2+b^2 )^(7/2)+2*a*b*(a^2-b^2)*cos(x)/(a^2+b^2)^3-2/3*a*b*cos(x)^3/(a^2+b^2)^2-b^ 2*(3*a^2-b^2)*sin(x)/(a^2+b^2)^3+1/3*(a^2-b^2)*sin(x)^3/(a^2+b^2)^2-a^2*b^ 3/(a^2+b^2)^3/(a*cos(x)+b*sin(x))
Time = 1.49 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 a b^2 \left (3 a^2-2 b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {-21 a^4 b+90 a^2 b^3-9 b^5-4 b \left (3 a^4+a^2 b^2-2 b^4\right ) \cos (2 x)+b \left (a^2+b^2\right )^2 \cos (4 x)-2 a^5 \sin (2 x)+16 a^3 b^2 \sin (2 x)+18 a b^4 \sin (2 x)+a^5 \sin (4 x)+2 a^3 b^2 \sin (4 x)+a b^4 \sin (4 x)}{24 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
(2*a*b^2*(3*a^2 - 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (-21*a^4*b + 90*a^2*b^3 - 9*b^5 - 4*b*(3*a^4 + a^2*b^2 - 2* b^4)*Cos[2*x] + b*(a^2 + b^2)^2*Cos[4*x] - 2*a^5*Sin[2*x] + 16*a^3*b^2*Sin [2*x] + 18*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^3*b^2*Sin[4*x] + a*b^4*Sin[ 4*x])/(24*(a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))
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.
| \(\displaystyle \int \frac {\sin ^2(x) \cos ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^2 \cos (x)^3}{(a \cos (x)+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle \frac {b \int \frac {\cos ^3(x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos ^2(x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^3 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \left (\frac {b \int \cos ^2(x) \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin ^2(x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \cos ^3(x)dx}{a^2+b^2}+\frac {a \int \cos ^2(x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \left (\frac {b \int \cos (x)^2 \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^3dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \left (\frac {a \int \sin ^2(x)d\sin (x)}{a^2+b^2}+\frac {b \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^3dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \left (\frac {b \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^3dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {a \left (-\frac {b \int \cos ^2(x)d\cos (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^3dx}{a^2+b^2}-\frac {a \int \cos ^2(x)d\cos (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^3dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {b \int \left (1-\sin ^2(x)\right )d(-\sin (x))}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (-\frac {a b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {a \int \cos (x)dx}{a^2+b^2}+\frac {b^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (\frac {a \int \sin \left (x+\frac {\pi }{2}\right )dx}{a^2+b^2}+\frac {b^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {b \left (-\frac {a b \left (\frac {b^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2+b^2}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \cos (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \sin \left (x+\frac {\pi }{2}\right )dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a \int \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {a \left (-\frac {a b \left (\frac {a b \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3590 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \frac {\cos ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \int \frac {\cos (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {a \int \cos (x)dx}{a^2+b^2}+\frac {b^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {a \int \sin \left (x+\frac {\pi }{2}\right )dx}{a^2+b^2}+\frac {b^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {b \left (\frac {b^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (-\frac {b^2 \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2+b^2}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{(a \cos (x)+b \sin (x))^2}dx}{a^2+b^2}+\frac {a \left (\frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \cos (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (-\frac {a b \left (-\frac {b^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \sin (x)}{a^2+b^2}+\frac {b \cos (x)}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )}{a^2+b^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}\) |
3.3.91.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*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]
Time = 1.02 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.51
| method | result | size |
| default | \(-\frac {2 \left (\left (3 a^{2} b^{2}-b^{4}\right ) \tan \left (\frac {x}{2}\right )^{5}+4 a \,b^{3} \tan \left (\frac {x}{2}\right )^{4}+\left (-\frac {4}{3} a^{4}+6 a^{2} b^{2}-\frac {2}{3} b^{4}\right ) \tan \left (\frac {x}{2}\right )^{3}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \tan \left (\frac {x}{2}\right )^{2}+\left (3 a^{2} b^{2}-b^{4}\right ) \tan \left (\frac {x}{2}\right )-\frac {4 a^{3} b}{3}+\frac {8 a \,b^{3}}{3}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}-\frac {2 a \,b^{2} \left (\frac {-b^{2} \tan \left (\frac {x}{2}\right )-a b}{\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a}-\frac {\left (3 a^{2}-2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}\) | \(265\) |
| risch | \(\frac {i {\mathrm e}^{3 i x}}{-48 i b a +24 a^{2}-24 b^{2}}+\frac {3 \,{\mathrm e}^{i x} b}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right )}-\frac {i {\mathrm e}^{i x} a}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right )}+\frac {3 \,{\mathrm e}^{-i x} b}{8 \left (i b +a \right )^{3}}+\frac {i {\mathrm e}^{-i x} a}{8 \left (i b +a \right )^{3}}-\frac {i {\mathrm e}^{-3 i x}}{24 \left (i b +a \right )^{2}}-\frac {2 a^{2} b^{3} {\mathrm e}^{i x}}{\left (i b +a \right )^{3} \left (-i b +a \right )^{3} \left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )}-\frac {3 b^{2} a^{3} \ln \left ({\mathrm e}^{i x}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {2 b^{4} a \ln \left ({\mathrm e}^{i x}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {3 b^{2} a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {2 b^{4} a \ln \left ({\mathrm e}^{i x}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\) | \(565\) |
-2/(a^2+b^2)/(a^4+2*a^2*b^2+b^4)*((3*a^2*b^2-b^4)*tan(1/2*x)^5+4*a*b^3*tan (1/2*x)^4+(-4/3*a^4+6*a^2*b^2-2/3*b^4)*tan(1/2*x)^3+(-4*a^3*b+4*a*b^3)*tan (1/2*x)^2+(3*a^2*b^2-b^4)*tan(1/2*x)-4/3*a^3*b+8/3*a*b^3)/(1+tan(1/2*x)^2) ^3-2*a*b^2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-b^2*tan(1/2*x)-a*b)/(tan(1/2*x )^2*a-2*b*tan(1/2*x)-a)-(3*a^2-2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan (1/2*x)-2*b)/(a^2+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (168) = 336\).
Time = 0.29 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.10 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 \, a^{6} b - 22 \, a^{4} b^{3} - 20 \, a^{2} b^{5} + 4 \, b^{7} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{4} + 2 \, {\left (4 \, a^{6} b + 7 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - b^{7}\right )} \cos \left (x\right )^{2} - 3 \, \sqrt {a^{2} + b^{2}} {\left ({\left (3 \, a^{4} b^{2} - 2 \, a^{2} b^{4}\right )} \cos \left (x\right ) + {\left (3 \, a^{3} b^{3} - 2 \, a b^{5}\right )} \sin \left (x\right )\right )} \log \left (\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )^{3} - {\left (a^{7} - 2 \, a^{5} b^{2} - 7 \, a^{3} b^{4} - 4 \, a b^{6}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{6 \, {\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (x\right ) + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \sin \left (x\right )\right )}} \]
1/6*(2*a^6*b - 22*a^4*b^3 - 20*a^2*b^5 + 4*b^7 - 2*(a^6*b + 3*a^4*b^3 + 3* a^2*b^5 + b^7)*cos(x)^4 + 2*(4*a^6*b + 7*a^4*b^3 + 2*a^2*b^5 - b^7)*cos(x) ^2 - 3*sqrt(a^2 + b^2)*((3*a^4*b^2 - 2*a^2*b^4)*cos(x) + (3*a^3*b^3 - 2*a* b^5)*sin(x))*log((2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 - 2*a^2 - b^2 - 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)) - 2*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(x)^3 - (a^7 - 2*a^5*b^2 - 7*a^3*b^4 - 4*a*b^6)*cos(x))*sin(x))/((a^9 + 4*a^7*b^ 2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cos(x) + (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*sin(x))
Timed out. \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (168) = 336\).
Time = 0.33 (sec) , antiderivative size = 606, normalized size of antiderivative = 3.44 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {{\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} a \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (4 \, a^{4} b - 11 \, a^{2} b^{3} - \frac {{\left (a^{3} b^{2} + 16 \, a b^{4}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {{\left (8 \, a^{4} b - 31 \, a^{2} b^{3} + 6 \, b^{5}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (4 \, a^{5} + 15 \, a^{3} b^{2} - 34 \, a b^{4}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (4 \, a^{4} b + 45 \, a^{2} b^{3} - 4 \, b^{5}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {{\left (4 \, a^{5} - 9 \, a^{3} b^{2} + 32 \, a b^{4}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {3 \, {\left (3 \, a^{2} b^{3} - 2 \, b^{5}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {3 \, {\left (3 \, a^{3} b^{2} - 2 \, a b^{4}\right )} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}}{3 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} + \frac {2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {6 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac {{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} \]
-(3*a^2*b^2 - 2*b^4)*a*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/( b - a*sin(x)/(cos(x) + 1) - sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^ 4 + b^6)*sqrt(a^2 + b^2)) + 2/3*(4*a^4*b - 11*a^2*b^3 - (a^3*b^2 + 16*a*b^ 4)*sin(x)/(cos(x) + 1) + (8*a^4*b - 31*a^2*b^3 + 6*b^5)*sin(x)^2/(cos(x) + 1)^2 + (4*a^5 + 15*a^3*b^2 - 34*a*b^4)*sin(x)^3/(cos(x) + 1)^3 - (4*a^4*b + 45*a^2*b^3 - 4*b^5)*sin(x)^4/(cos(x) + 1)^4 - (4*a^5 - 9*a^3*b^2 + 32*a *b^4)*sin(x)^5/(cos(x) + 1)^5 - 3*(3*a^2*b^3 - 2*b^5)*sin(x)^6/(cos(x) + 1 )^6 + 3*(3*a^3*b^2 - 2*a*b^4)*sin(x)^7/(cos(x) + 1)^7)/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sin(x)/(cos(x) + 1) + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*sin(x)^2/(cos(x) + 1)^2 + 6*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sin(x)^3/(cos(x) + 1)^3 + 6*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sin(x)^5/(cos(x) + 1)^5 - 2*(a^7 + 3*a^5*b ^2 + 3*a^3*b^4 + a*b^6)*sin(x)^6/(cos(x) + 1)^6 + 2*(a^6*b + 3*a^4*b^3 + 3 *a^2*b^5 + b^7)*sin(x)^7/(cos(x) + 1)^7 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a *b^6)*sin(x)^8/(cos(x) + 1)^8)
Time = 0.31 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {{\left (3 \, a^{3} b^{2} - 2 \, a b^{4}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{4} \tan \left (\frac {1}{2} \, x\right ) + a^{2} b^{3}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )}} - \frac {2 \, {\left (9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{5} - 3 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{5} + 12 \, a b^{3} \tan \left (\frac {1}{2} \, x\right )^{4} - 4 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} - 2 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{3} - 12 \, a^{3} b \tan \left (\frac {1}{2} \, x\right )^{2} + 12 \, a b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right ) - 3 \, b^{4} \tan \left (\frac {1}{2} \, x\right ) - 4 \, a^{3} b + 8 \, a b^{3}\right )}}{3 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} \]
-(3*a^3*b^2 - 2*a*b^4)*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/a bs(2*a*tan(1/2*x) - 2*b + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^ 4 + b^6)*sqrt(a^2 + b^2)) + 2*(a*b^4*tan(1/2*x) + a^2*b^3)/((a^6 + 3*a^4*b ^2 + 3*a^2*b^4 + b^6)*(a*tan(1/2*x)^2 - 2*b*tan(1/2*x) - a)) - 2/3*(9*a^2* b^2*tan(1/2*x)^5 - 3*b^4*tan(1/2*x)^5 + 12*a*b^3*tan(1/2*x)^4 - 4*a^4*tan( 1/2*x)^3 + 18*a^2*b^2*tan(1/2*x)^3 - 2*b^4*tan(1/2*x)^3 - 12*a^3*b*tan(1/2 *x)^2 + 12*a*b^3*tan(1/2*x)^2 + 9*a^2*b^2*tan(1/2*x) - 3*b^4*tan(1/2*x) - 4*a^3*b + 8*a*b^3)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(1/2*x)^2 + 1) ^3)
Time = 24.91 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.33 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (4\,a^4\,b+45\,a^2\,b^3-4\,b^5\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6\,\left (2\,b^5-3\,a^2\,b^3\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\left (4\,a^5-9\,a^3\,b^2+32\,a\,b^4\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (4\,a^5+15\,a^3\,b^2-34\,a\,b^4\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (8\,a^4\,b-31\,a^2\,b^3+6\,b^5\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,\left (11\,a\,b^3-4\,a^3\,b\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7\,\left (2\,a\,b^3-3\,a^3\,b\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^3\,b+16\,a\,b^3\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7-2\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+6\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+6\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a}-\frac {a\,b^2\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^7-a^6\,b\,1{}\mathrm {i}+3{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^5\,b^2-a^4\,b^3\,3{}\mathrm {i}+3{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^3\,b^4-a^2\,b^5\,3{}\mathrm {i}+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {x}{2}\right )\,a\,b^6-b^7\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{7/2}}\right )\,\left (3\,a^2-2\,b^2\right )\,2{}\mathrm {i}}{{\left (a^2+b^2\right )}^{7/2}} \]
- ((2*tan(x/2)^4*(4*a^4*b - 4*b^5 + 45*a^2*b^3))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (2*tan(x/2)^6*(2*b^5 - 3*a^2*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*tan(x/2)^5*(32*a*b^4 + 4*a^5 - 9*a^3*b^2))/(3*(a^2 + b^ 2)*(a^4 + b^4 + 2*a^2*b^2)) - (2*tan(x/2)^3*(4*a^5 - 34*a*b^4 + 15*a^3*b^2 ))/(3*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) - (2*tan(x/2)^2*(8*a^4*b + 6*b^ 5 - 31*a^2*b^3))/(3*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) + (2*a*(11*a*b^3 - 4*a^3*b))/(3*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) + (2*b*tan(x/2)^7*(2*a *b^3 - 3*a^3*b))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*b*tan(x/2)*(16*a *b^3 + a^3*b))/(3*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)))/(a + 2*b*tan(x/2) + 2*a*tan(x/2)^2 - 2*a*tan(x/2)^6 - a*tan(x/2)^8 + 6*b*tan(x/2)^3 + 6*b*ta n(x/2)^5 + 2*b*tan(x/2)^7) - (a*b^2*atan((a^7*tan(x/2)*1i - a^6*b*1i - b^7 *1i - a^2*b^5*3i - a^4*b^3*3i + a^3*b^4*tan(x/2)*3i + a^5*b^2*tan(x/2)*3i + a*b^6*tan(x/2)*1i)/(a^2 + b^2)^(7/2))*(3*a^2 - 2*b^2)*2i)/(a^2 + b^2)^(7 /2)