Integrand size = 27, antiderivative size = 355 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {15 b \left (a^2-2 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}-\frac {15 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}+\frac {\left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{2 a^6 b d}+\frac {15 \left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {\left (7 a^2-10 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))} \] Output:
-15*b*(a^2-2*b^2)*(a^2-b^2)^(1/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2 )^(1/2))/a^7/d-15/8*(a^4-8*a^2*b^2+8*b^4)*arctanh(cos(d*x+c))/a^7/d+1/2*(a ^4-25*a^2*b^2+30*b^4)*cot(d*x+c)/a^6/b/d+15/8*(3*a^2-4*b^2)*cot(d*x+c)*csc (d*x+c)/a^5/d-1/2*cot(d*x+c)/b/d/(a+b*sin(d*x+c))^2-1/4*(4*a^2-5*b^2)*cot( d*x+c)*csc(d*x+c)/a^3/d/(a+b*sin(d*x+c))^2+1/2*b*cot(d*x+c)*csc(d*x+c)^2/a ^2/d/(a+b*sin(d*x+c))^2-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d/(a+b*sin(d*x+c))^2 -1/2*(7*a^2-10*b^2)*cot(d*x+c)*csc(d*x+c)/a^4/d/(a+b*sin(d*x+c))
Time = 2.01 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.02 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {1920 b \left (a^4-3 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-240 \left (a^4-8 a^2 b^2+8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+240 \left (a^4-8 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^5(c+d x) \left (44 a^5-505 a^3 b^2+540 a b^4+\left (-68 a^5+660 a^3 b^2-720 a b^4\right ) \cos (2 (c+d x))+\left (8 a^5-155 a^3 b^2+180 a b^4\right ) \cos (4 (c+d x))-176 a^4 b \sin (c+d x)-260 a^2 b^3 \sin (c+d x)+600 b^5 \sin (c+d x)+66 a^4 b \sin (3 (c+d x))+170 a^2 b^3 \sin (3 (c+d x))-300 b^5 \sin (3 (c+d x))+2 a^4 b \sin (5 (c+d x))-50 a^2 b^3 \sin (5 (c+d x))+60 b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{128 a^7 d} \] Input:
Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^3,x]
Output:
((-1920*b*(a^4 - 3*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a ^2 - b^2]])/Sqrt[a^2 - b^2] - 240*(a^4 - 8*a^2*b^2 + 8*b^4)*Log[Cos[(c + d *x)/2]] + 240*(a^4 - 8*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^5*(44*a^5 - 505*a^3*b^2 + 540*a*b^4 + (-68*a^5 + 660* a^3*b^2 - 720*a*b^4)*Cos[2*(c + d*x)] + (8*a^5 - 155*a^3*b^2 + 180*a*b^4)* Cos[4*(c + d*x)] - 176*a^4*b*Sin[c + d*x] - 260*a^2*b^3*Sin[c + d*x] + 600 *b^5*Sin[c + d*x] + 66*a^4*b*Sin[3*(c + d*x)] + 170*a^2*b^3*Sin[3*(c + d*x )] - 300*b^5*Sin[3*(c + d*x)] + 2*a^4*b*Sin[5*(c + d*x)] - 50*a^2*b^3*Sin[ 5*(c + d*x)] + 60*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(128*a^7* d)
Time = 3.29 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.30, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {3042, 3375, 27, 3042, 3534, 27, 3042, 3534, 3042, 3534, 3042, 3534, 27, 3042, 3480, 3042, 3139, 1083, 217, 4257}
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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^5 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int -\frac {18 \csc ^3(c+d x) \left (-2 \left (3 a^2-4 b^2\right ) \sin ^2(c+d x) b^2+\left (9 a^2-10 b^2\right ) b^2+a \left (2 a^2-b^2\right ) \sin (c+d x) b\right )}{(a+b \sin (c+d x))^3}dx}{72 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\csc ^3(c+d x) \left (-2 \left (3 a^2-4 b^2\right ) \sin ^2(c+d x) b^2+\left (9 a^2-10 b^2\right ) b^2+a \left (2 a^2-b^2\right ) \sin (c+d x) b\right )}{(a+b \sin (c+d x))^3}dx}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-2 \left (3 a^2-4 b^2\right ) \sin (c+d x)^2 b^2+\left (9 a^2-10 b^2\right ) b^2+a \left (2 a^2-b^2\right ) \sin (c+d x) b}{\sin (c+d x)^3 (a+b \sin (c+d x))^3}dx}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int \frac {2 \csc ^3(c+d x) \left (-3 \left (4 a^4-9 b^2 a^2+5 b^4\right ) \sin ^2(c+d x) b^2+\left (17 a^4-37 b^2 a^2+20 b^4\right ) b^2+2 a \left (a^2-b^2\right )^2 \sin (c+d x) b\right )}{(a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^3(c+d x) \left (-3 \left (4 a^4-9 b^2 a^2+5 b^4\right ) \sin ^2(c+d x) b^2+\left (17 a^4-37 b^2 a^2+20 b^4\right ) b^2+2 a \left (a^2-b^2\right )^2 \sin (c+d x) b\right )}{(a+b \sin (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {-3 \left (4 a^4-9 b^2 a^2+5 b^4\right ) \sin (c+d x)^2 b^2+\left (17 a^4-37 b^2 a^2+20 b^4\right ) b^2+2 a \left (a^2-b^2\right )^2 \sin (c+d x) b}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\csc ^3(c+d x) \left (-4 b^2 \left (7 a^2-10 b^2\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2+a b \left (2 a^2-5 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {-4 b^2 \left (7 a^2-10 b^2\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2+a b \left (2 a^2-5 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\csc ^2(c+d x) \left (15 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x) b^3-a \left (11 a^2-20 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^2+4 \left (a^2-b^2\right )^2 \left (a^4-25 b^2 a^2+30 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {15 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)^2 b^3-a \left (11 a^2-20 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^2+4 \left (a^2-b^2\right )^2 \left (a^4-25 b^2 a^2+30 b^4\right ) b}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {\int -\frac {15 \csc (c+d x) \left (b^2 \left (a^2-b^2\right )^2 \left (a^4-8 b^2 a^2+8 b^4\right )-a b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {4 b \left (a^2-b^2\right )^2 \left (a^4-25 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \int \frac {\csc (c+d x) \left (b^2 \left (a^2-b^2\right )^2 \left (a^4-8 b^2 a^2+8 b^4\right )-a b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \int \frac {b^2 \left (a^2-b^2\right )^2 \left (a^4-8 b^2 a^2+8 b^4\right )-a b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \left (\frac {b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {4 b^3 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \left (\frac {b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {4 b^3 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \left (\frac {b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^3 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \left (\frac {16 b^3 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b^2 \left (a^4-8 a^2 b^2+8 b^4\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}\right )}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {15 \left (\frac {b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^3 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}}{4 a^2 b^2}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac {\frac {b^2 \left (4 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))^2}+\frac {\frac {2 b^2 \left (7 a^2-10 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\frac {-\frac {15 \left (-\frac {8 b^3 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^2 \left (a^2-b^2\right )^2 \left (a^4-8 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {4 b \left (a^4-25 a^2 b^2+30 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}}{a \left (a^2-b^2\right )}}{4 a^2 b^2}-\frac {\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
Input:
Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^3,x]
Output:
-1/2*Cot[c + d*x]/(b*d*(a + b*Sin[c + d*x])^2) + (b*Cot[c + d*x]*Csc[c + d *x]^2)/(2*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4 *a*d*(a + b*Sin[c + d*x])^2) - ((b^2*(4*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a*d*(a + b*Sin[c + d*x])^2) + ((((-15*((-8*b^3*(a^2 - 2*b^2)*(a^2 - b^2)^(5/2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d ) - (b^2*(a^2 - b^2)^2*(a^4 - 8*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(a *d)))/a - (4*b*(a^2 - b^2)^2*(a^4 - 25*a^2*b^2 + 30*b^4)*Cot[c + d*x])/(a* d))/(2*a) - (15*b^2*(3*a^2 - 4*b^2)*(a^2 - b^2)^2*Cot[c + d*x]*Csc[c + d*x ])/(2*a*d))/(a*(a^2 - b^2)) + (2*b^2*(7*a^2 - 10*b^2)*(a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x])/(a*d*(a + b*Sin[c + d*x])))/(a*(a^2 - b^2)))/(4*a^2*b^2 )
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x] )^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f*x]*(d *Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5) *(m + n + 6))), x] + Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin [e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x] + Simp[1/(a^2*b^2*d^2*(n + 1) *(n + 2)*(m + n + 5)*(m + n + 6)) Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin [e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2* n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4 )*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && Ne Q[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && !IGtQ[m, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 4.50 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+12 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+54 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-240 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (27 a^{2}-40 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \left (\frac {\left (-\frac {3}{2} a^{5} b +\frac {15}{2} a^{3} b^{3}-6 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{6}+\frac {9}{2} a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-11 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a b \left (5 a^{4}-37 a^{2} b^{2}+32 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{2} \left (2 a^{4}-13 a^{2} b^{2}+11 b^{4}\right )}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {15 b \left (a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}}{d}\) | \(455\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+12 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+54 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-240 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (27 a^{2}-40 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \left (\frac {\left (-\frac {3}{2} a^{5} b +\frac {15}{2} a^{3} b^{3}-6 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{6}+\frac {9}{2} a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-11 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a b \left (5 a^{4}-37 a^{2} b^{2}+32 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{2} \left (2 a^{4}-13 a^{2} b^{2}+11 b^{4}\right )}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {15 b \left (a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}}{d}\) | \(455\) |
| risch | \(\text {Expression too large to display}\) | \(1022\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/16/a^6*(1/4*tan(1/2*d*x+1/2*c)^4*a^3-2*b*tan(1/2*d*x+1/2*c)^3*a^2-4 *tan(1/2*d*x+1/2*c)^2*a^3+12*a*b^2*tan(1/2*d*x+1/2*c)^2+54*a^2*b*tan(1/2*d *x+1/2*c)-80*tan(1/2*d*x+1/2*c)*b^3)-1/64/a^3/tan(1/2*d*x+1/2*c)^4-1/32*(- 8*a^2+24*b^2)/a^5/tan(1/2*d*x+1/2*c)^2+1/16/a^7*(30*a^4-240*a^2*b^2+240*b^ 4)*ln(tan(1/2*d*x+1/2*c))+1/8/a^4*b/tan(1/2*d*x+1/2*c)^3-1/8*b*(27*a^2-40* b^2)/a^6/tan(1/2*d*x+1/2*c)-2/a^7*(((-3/2*a^5*b+15/2*a^3*b^3-6*a*b^5)*tan( 1/2*d*x+1/2*c)^3+(-a^6+9/2*a^4*b^2+15/2*a^2*b^4-11*b^6)*tan(1/2*d*x+1/2*c) ^2-1/2*a*b*(5*a^4-37*a^2*b^2+32*b^4)*tan(1/2*d*x+1/2*c)-1/2*a^2*(2*a^4-13* a^2*b^2+11*b^4))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+15/2* b*(a^4-3*a^2*b^2+2*b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c) +2*b)/(a^2-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (334) = 668\).
Time = 0.40 (sec) , antiderivative size = 2022, normalized size of antiderivative = 5.70 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
[-1/16*(2*(8*a^6 - 155*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 10*(5*a^6 - 64*a^4*b^2 + 72*a^2*b^4)*cos(d*x + c)^3 + 60*((a^2*b^3 - 2*b^5)*cos(d*x + c)^6 - a^4*b + a^2*b^3 + 2*b^5 - (a^4*b + a^2*b^3 - 6*b^5)*cos(d*x + c)^4 + (2*a^4*b - a^2*b^3 - 6*b^5)*cos(d*x + c)^2 - 2*(a^3*b^2 - 2*a*b^4 + (a^ 3*b^2 - 2*a*b^4)*cos(d*x + c)^4 - 2*(a^3*b^2 - 2*a*b^4)*cos(d*x + c)^2)*si n(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*si n(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))* sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 30*(a^6 - 11*a^4*b^2 + 12*a^2*b^4)*cos(d*x + c) + 15*((a^4*b^2 - 8*a^2*b^ 4 + 8*b^6)*cos(d*x + c)^6 - a^6 + 7*a^4*b^2 - 8*b^6 - (a^6 - 5*a^4*b^2 - 1 6*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + (2*a^6 - 13*a^4*b^2 - 8*a^2*b^4 + 24* b^6)*cos(d*x + c)^2 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5 + (a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^2 )*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*((a^4*b^2 - 8*a^2*b^4 + 8 *b^6)*cos(d*x + c)^6 - a^6 + 7*a^4*b^2 - 8*b^6 - (a^6 - 5*a^4*b^2 - 16*a^2 *b^4 + 24*b^6)*cos(d*x + c)^4 + (2*a^6 - 13*a^4*b^2 - 8*a^2*b^4 + 24*b^6)* cos(d*x + c)^2 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5 + (a^5*b - 8*a^3*b^3 + 8*a *b^5)*cos(d*x + c)^4 - 2*(a^5*b - 8*a^3*b^3 + 8*a*b^5)*cos(d*x + c)^2)*sin (d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 4*(2*(a^5*b - 25*a^3*b^3 + 30*a* b^5)*cos(d*x + c)^5 + 5*(3*a^5*b + 16*a^3*b^3 - 24*a*b^5)*cos(d*x + c)^...
\[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**5/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)*cot(c + d*x)**5/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.28 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.70 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/64*(120*(a^4 - 8*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 - 9 60*(a^4*b - 3*a^2*b^3 + 2*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^7 ) + 64*(3*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 2*a^6*tan(1/2*d*x + 1/2*c)^2 - 9*a^4* b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 22*b^6*ta n(1/2*d*x + 1/2*c)^2 + 5*a^5*b*tan(1/2*d*x + 1/2*c) - 37*a^3*b^3*tan(1/2*d *x + 1/2*c) + 32*a*b^5*tan(1/2*d*x + 1/2*c) + 2*a^6 - 13*a^4*b^2 + 11*a^2* b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^7) - ( 250*a^4*tan(1/2*d*x + 1/2*c)^4 - 2000*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 200 0*b^4*tan(1/2*d*x + 1/2*c)^4 + 216*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^ 3*tan(1/2*d*x + 1/2*c)^3 - 16*a^4*tan(1/2*d*x + 1/2*c)^2 + 48*a^2*b^2*tan( 1/2*d*x + 1/2*c)^2 - 8*a^3*b*tan(1/2*d*x + 1/2*c) + a^4)/(a^7*tan(1/2*d*x + 1/2*c)^4) + (a^9*tan(1/2*d*x + 1/2*c)^4 - 8*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 16*a^9*tan(1/2*d*x + 1/2*c)^2 + 48*a^7*b^2*tan(1/2*d*x + 1/2*c)^2 + 216 *a^8*b*tan(1/2*d*x + 1/2*c) - 320*a^6*b^3*tan(1/2*d*x + 1/2*c))/a^12)/d
Time = 23.43 (sec) , antiderivative size = 1275, normalized size of antiderivative = 3.59 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)*cot(c + d*x)^5)/(a + b*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)^4/(64*a^3*d) - (tan(c/2 + (d*x)/2)^3*(35*a^4*b - 40*a^2 *b^3) - tan(c/2 + (d*x)/2)^4*(448*a*b^4 + (159*a^5)/4 - 424*a^3*b^2) + tan (c/2 + (d*x)/2)^7*(6*a^4*b - 192*b^5 + 160*a^2*b^3) + tan(c/2 + (d*x)/2)^5 *(10*a^4*b - 832*b^5 + 696*a^2*b^3) + a^5/4 - tan(c/2 + (d*x)/2)^2*((7*a^5 )/2 - 5*a^3*b^2) - a^4*b*tan(c/2 + (d*x)/2) - (4*tan(c/2 + (d*x)/2)^6*(9*a ^6 + 88*b^6 + 20*a^2*b^4 - 93*a^4*b^2))/a)/(d*(16*a^8*tan(c/2 + (d*x)/2)^4 + 16*a^8*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^6*(32*a^8 + 64*a^6*b^2 ) + 64*a^7*b*tan(c/2 + (d*x)/2)^5 + 64*a^7*b*tan(c/2 + (d*x)/2)^7)) - (tan (c/2 + (d*x)/2)^2*((3*(a^2 + 4*b^2))/(32*a^5) + 5/(32*a^3) - (9*b^2)/(8*a^ 5)))/d + (tan(c/2 + (d*x)/2)*((6*b*((3*(a^2 + 4*b^2))/(16*a^5) + 5/(16*a^3 ) - (9*b^2)/(4*a^5)))/a - (192*a^2*b + 128*b^3)/(256*a^6) + (9*b*(a^2 + 4* b^2))/(8*a^6)))/d - (b*tan(c/2 + (d*x)/2)^3)/(8*a^4*d) + (log(tan(c/2 + (d *x)/2))*((15*a^4)/8 + 15*b^4 - 15*a^2*b^2))/(a^7*d) + (b*atan(((b*(b^2 - a ^2)^(1/2)*(a^2 - 2*b^2)*(((75*a^11*b)/4 + 60*a^7*b^5 - 75*a^9*b^3)/a^12 - (tan(c/2 + (d*x)/2)*(15*a^11 - 480*a^5*b^6 + 720*a^7*b^4 - 270*a^9*b^2))/( 4*a^11) + (15*b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^14 - 32*a^12*b^2))/(4 *a^11))*(b^2 - a^2)^(1/2)*(a^2 - 2*b^2))/(2*a^7))*15i)/(2*a^7) - (b*(b^2 - a^2)^(1/2)*(a^2 - 2*b^2)*((tan(c/2 + (d*x)/2)*(15*a^11 - 480*a^5*b^6 + 72 0*a^7*b^4 - 270*a^9*b^2))/(4*a^11) - ((75*a^11*b)/4 + 60*a^7*b^5 - 75*a^9* b^3)/a^12 + (15*b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^14 - 32*a^12*b^2...
Time = 0.20 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.68 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)*cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x)
Output:
( - 240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2)) *sin(c + d*x)**6*a**2*b**3 + 480*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)* a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**6*b**5 - 480*sqrt(a**2 - b**2)*ata n((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**3*b**2 + 960*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin (c + d*x)**5*a*b**4 - 240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/ sqrt(a**2 - b**2))*sin(c + d*x)**4*a**4*b + 480*sqrt(a**2 - b**2)*atan((ta n((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**2*b**3 + 8*cos (c + d*x)*sin(c + d*x)**5*a**5*b - 200*cos(c + d*x)*sin(c + d*x)**5*a**3*b **3 + 240*cos(c + d*x)*sin(c + d*x)**5*a*b**5 + 16*cos(c + d*x)*sin(c + d* x)**4*a**6 - 310*cos(c + d*x)*sin(c + d*x)**4*a**4*b**2 + 360*cos(c + d*x) *sin(c + d*x)**4*a**2*b**4 - 76*cos(c + d*x)*sin(c + d*x)**3*a**5*b + 80*c os(c + d*x)*sin(c + d*x)**3*a**3*b**3 + 18*cos(c + d*x)*sin(c + d*x)**2*a* *6 - 20*cos(c + d*x)*sin(c + d*x)**2*a**4*b**2 + 8*cos(c + d*x)*sin(c + d* x)*a**5*b - 4*cos(c + d*x)*a**6 + 30*log(tan((c + d*x)/2))*sin(c + d*x)**6 *a**4*b**2 - 240*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**2*b**4 + 240*log (tan((c + d*x)/2))*sin(c + d*x)**6*b**6 + 60*log(tan((c + d*x)/2))*sin(c + d*x)**5*a**5*b - 480*log(tan((c + d*x)/2))*sin(c + d*x)**5*a**3*b**3 + 48 0*log(tan((c + d*x)/2))*sin(c + d*x)**5*a*b**5 + 30*log(tan((c + d*x)/2))* sin(c + d*x)**4*a**6 - 240*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4*b...