Integrand size = 27, antiderivative size = 340 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 \sqrt {a^2-b^2} d}-\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^7 d}-\frac {b \left (13 a^2-30 b^2\right ) \cot (c+d x)}{2 a^6 d}+\frac {3 \left (7 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac {\left (3 a^2-10 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))} \] Output:
-3*b*(2*a^4-11*a^2*b^2+10*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^( 1/2))/a^7/(a^2-b^2)^(1/2)/d-3/8*(a^4-24*a^2*b^2+40*b^4)*arctanh(cos(d*x+c) )/a^7/d-1/2*b*(13*a^2-30*b^2)*cot(d*x+c)/a^6/d+3/8*(7*a^2-20*b^2)*cot(d*x+ c)*csc(d*x+c)/a^5/d-1/2*(3*a^2-10*b^2)*cot(d*x+c)*csc(d*x+c)^2/a^4/b/d+1/4 *(2*a^2-3*b^2)*cot(d*x+c)*csc(d*x+c)^2/a^2/b/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+1/4*(4*a^2-15*b^2)*cot(d*x+c)*c sc(d*x+c)^2/a^3/b/d/(a+b*sin(d*x+c))
Time = 6.51 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {384 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+48 \left (a^4-24 a^2 b^2+40 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-48 \left (a^4-24 a^2 b^2+40 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^5(c+d x) \left (-4 a^5+289 a^3 b^2-540 a b^4+4 \left (5 a^5-93 a^3 b^2+180 a b^4\right ) \cos (2 (c+d x))+\left (83 a^3 b^2-180 a b^4\right ) \cos (4 (c+d x))+100 a^4 b \sin (c+d x)+20 a^2 b^3 \sin (c+d x)-600 b^5 \sin (c+d x)-44 a^4 b \sin (3 (c+d x))-50 a^2 b^3 \sin (3 (c+d x))+300 b^5 \sin (3 (c+d x))+26 a^2 b^3 \sin (5 (c+d x))-60 b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{128 a^7 d} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]
Output:
-1/128*((384*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2 ])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 48*(a^4 - 24*a^2*b^2 + 40*b^4)*Log[ Cos[(c + d*x)/2]] - 48*(a^4 - 24*a^2*b^2 + 40*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^5*(-4*a^5 + 289*a^3*b^2 - 540*a*b^4 + 4*(5 *a^5 - 93*a^3*b^2 + 180*a*b^4)*Cos[2*(c + d*x)] + (83*a^3*b^2 - 180*a*b^4) *Cos[4*(c + d*x)] + 100*a^4*b*Sin[c + d*x] + 20*a^2*b^3*Sin[c + d*x] - 600 *b^5*Sin[c + d*x] - 44*a^4*b*Sin[3*(c + d*x)] - 50*a^2*b^3*Sin[3*(c + d*x) ] + 300*b^5*Sin[3*(c + d*x)] + 26*a^2*b^3*Sin[5*(c + d*x)] - 60*b^5*Sin[5* (c + d*x)]))/(b + a*Csc[c + d*x])^2)/(a^7*d)
Time = 2.94 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.26, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {3042, 3369, 27, 3042, 3534, 27, 3042, 3534, 25, 3042, 3534, 25, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^5 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3369 |
| \(\displaystyle \frac {\int \frac {2 \csc ^4(c+d x) \left (-4 \left (a^2-3 b^2\right ) \sin ^2(c+d x)-a b \sin (c+d x)+3 \left (2 a^2-5 b^2\right )\right )}{(a+b \sin (c+d x))^2}dx}{8 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x) \left (-4 \left (a^2-3 b^2\right ) \sin ^2(c+d x)-a b \sin (c+d x)+3 \left (2 a^2-5 b^2\right )\right )}{(a+b \sin (c+d x))^2}dx}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-4 \left (a^2-3 b^2\right ) \sin (c+d x)^2-a b \sin (c+d x)+3 \left (2 a^2-5 b^2\right )}{\sin (c+d x)^4 (a+b \sin (c+d x))^2}dx}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {3 \csc ^4(c+d x) \left (-\left (\left (4 a^4-19 b^2 a^2+15 b^4\right ) \sin ^2(c+d x)\right )-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (3 a^4-13 b^2 a^2+10 b^4\right )\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {\csc ^4(c+d x) \left (-\left (\left (4 a^4-19 b^2 a^2+15 b^4\right ) \sin ^2(c+d x)\right )-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (3 a^4-13 b^2 a^2+10 b^4\right )\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \int \frac {-\left (\left (4 a^4-19 b^2 a^2+15 b^4\right ) \sin (c+d x)^2\right )-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (3 a^4-13 b^2 a^2+10 b^4\right )}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int -\frac {\csc ^3(c+d x) \left (-5 a \left (a^2-b^2\right ) \sin (c+d x) b^2-4 \left (3 a^4-13 b^2 a^2+10 b^4\right ) \sin ^2(c+d x) b+3 \left (7 a^4-27 b^2 a^2+20 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {\csc ^3(c+d x) \left (-5 a \left (a^2-b^2\right ) \sin (c+d x) b^2-4 \left (3 a^4-13 b^2 a^2+10 b^4\right ) \sin ^2(c+d x) b+3 \left (7 a^4-27 b^2 a^2+20 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {-5 a \left (a^2-b^2\right ) \sin (c+d x) b^2-4 \left (3 a^4-13 b^2 a^2+10 b^4\right ) \sin (c+d x)^2 b+3 \left (7 a^4-27 b^2 a^2+20 b^4\right ) b}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-3 \left (7 a^4-27 b^2 a^2+20 b^4\right ) \sin ^2(c+d x) b^2+4 \left (13 a^4-43 b^2 a^2+30 b^4\right ) b^2+a \left (3 a^4-23 b^2 a^2+20 b^4\right ) \sin (c+d x) b\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-3 \left (7 a^4-27 b^2 a^2+20 b^4\right ) \sin ^2(c+d x) b^2+4 \left (13 a^4-43 b^2 a^2+30 b^4\right ) b^2+a \left (3 a^4-23 b^2 a^2+20 b^4\right ) \sin (c+d x) b\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\int \frac {-3 \left (7 a^4-27 b^2 a^2+20 b^4\right ) \sin (c+d x)^2 b^2+4 \left (13 a^4-43 b^2 a^2+30 b^4\right ) b^2+a \left (3 a^4-23 b^2 a^2+20 b^4\right ) \sin (c+d x) b}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {\int \frac {3 \csc (c+d x) \left (b \left (a^6-25 b^2 a^4+64 b^4 a^2-40 b^6\right )-a b^2 \left (7 a^4-27 b^2 a^2+20 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \int \frac {\csc (c+d x) \left (b \left (a^6-25 b^2 a^4+64 b^4 a^2-40 b^6\right )-a b^2 \left (7 a^4-27 b^2 a^2+20 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \int \frac {b \left (a^6-25 b^2 a^4+64 b^4 a^2-40 b^6\right )-a b^2 \left (7 a^4-27 b^2 a^2+20 b^4\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \left (\frac {b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {4 b^2 \left (2 a^6-13 a^4 b^2+21 a^2 b^4-10 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \left (\frac {b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {4 b^2 \left (2 a^6-13 a^4 b^2+21 a^2 b^4-10 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \left (\frac {b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^2 \left (2 a^6-13 a^4 b^2+21 a^2 b^4-10 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \left (\frac {16 b^2 \left (2 a^6-13 a^4 b^2+21 a^2 b^4-10 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {-\frac {\frac {3 \left (\frac {b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^2 \left (2 a^6-13 a^4 b^2+21 a^2 b^4-10 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}\right )}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{4 a^2 b}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}+\frac {\frac {\left (4 a^2-15 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac {3 \left (-\frac {2 \left (3 a^4-13 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}-\frac {-\frac {3 b \left (7 a^4-27 a^2 b^2+20 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\frac {3 \left (-\frac {8 b^2 \left (2 a^6-13 a^4 b^2+21 a^2 b^4-10 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {b \left (a^6-25 a^4 b^2+64 a^2 b^4-40 b^6\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {4 b^2 \left (13 a^4-43 a^2 b^2+30 b^4\right ) \cot (c+d x)}{a d}}{2 a}}{3 a}\right )}{a \left (a^2-b^2\right )}}{4 a^2 b}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]
Output:
((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*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) + (( 3*((-2*(3*a^4 - 13*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d) - (-1/2*((3*((-8*b^2*(2*a^6 - 13*a^4*b^2 + 21*a^2*b^4 - 10*b^6)*ArcTan[(2* b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (b *(a^6 - 25*a^4*b^2 + 64*a^2*b^4 - 40*b^6)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (4*b^2*(13*a^4 - 43*a^2*b^2 + 30*b^4)*Cot[c + d*x])/(a*d))/a - (3*b*(7* a^4 - 27*a^2*b^2 + 20*b^4)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a)))/(a* (a^2 - b^2)) + ((4*a^2 - 15*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(a*d*(a + b* Sin[c + d*x])))/(4*a^2*b)
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_)]^4*((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[(a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(( a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1))), x] + Simp[1/(a^ 2*b*d*(n + 1)*(m + 1)) Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^ (m + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1 )*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]
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 = 2.92 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+12 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {2 b \left (\frac {\left (\frac {7}{2} a^{3} b^{2}-6 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {b \left (6 a^{4}+a^{2} b^{2}-22 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \,b^{2} \left (17 a^{2}-32 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (6 a^{2}-11 b^{2}\right )}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-144 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 b \left (3 a^{2}-8 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(431\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+12 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{6}}-\frac {2 b \left (\frac {\left (\frac {7}{2} a^{3} b^{2}-6 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {b \left (6 a^{4}+a^{2} b^{2}-22 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \,b^{2} \left (17 a^{2}-32 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (6 a^{2}-11 b^{2}\right )}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7}}-\frac {1}{64 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+24 b^{2}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-144 a^{2} b^{2}+240 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{7}}+\frac {b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 b \left (3 a^{2}-8 b^{2}\right )}{8 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(431\) |
| risch | \(\text {Expression too large to display}\) | \(1128\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)/(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^2-2 *tan(1/2*d*x+1/2*c)^2*a^3+12*a*b^2*tan(1/2*d*x+1/2*c)^2+30*tan(1/2*d*x+1/2 *c)*a^2*b-80*tan(1/2*d*x+1/2*c)*b^3)-2*b/a^7*(((7/2*a^3*b^2-6*a*b^4)*tan(1 /2*d*x+1/2*c)^3+1/2*b*(6*a^4+a^2*b^2-22*b^4)*tan(1/2*d*x+1/2*c)^2+1/2*a*b^ 2*(17*a^2-32*b^2)*tan(1/2*d*x+1/2*c)+1/2*a^2*b*(6*a^2-11*b^2))/(tan(1/2*d* x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+3/2*(2*a^4-11*a^2*b^2+10*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)))-1/6 4/a^3/tan(1/2*d*x+1/2*c)^4-1/32*(-4*a^2+24*b^2)/a^5/tan(1/2*d*x+1/2*c)^2+1 /16/a^7*(6*a^4-144*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)^3-5/8*b*(3*a^2-8*b^2)/a^6/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 1254 vs. \(2 (321) = 642\).
Time = 0.58 (sec) , antiderivative size = 2592, normalized size of antiderivative = 7.62 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
[1/16*(2*(83*a^6*b^2 - 263*a^4*b^4 + 180*a^2*b^6)*cos(d*x + c)^5 + 2*(5*a^ 8 - 181*a^6*b^2 + 536*a^4*b^4 - 360*a^2*b^6)*cos(d*x + c)^3 + 12*(2*a^6*b - 9*a^4*b^3 - a^2*b^5 + 10*b^7 - (2*a^4*b^3 - 11*a^2*b^5 + 10*b^7)*cos(d*x + c)^6 + (2*a^6*b - 5*a^4*b^3 - 23*a^2*b^5 + 30*b^7)*cos(d*x + c)^4 - (4* a^6*b - 16*a^4*b^3 - 13*a^2*b^5 + 30*b^7)*cos(d*x + c)^2 + 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6 + (2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6)*cos(d*x + c)^4 - 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*sqr t(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(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)) - 6*(a^8 - 32*a^ 6*b^2 + 91*a^4*b^4 - 60*a^2*b^6)*cos(d*x + c) + 3*(a^8 - 24*a^6*b^2 + 39*a ^4*b^4 + 24*a^2*b^6 - 40*b^8 - (a^6*b^2 - 25*a^4*b^4 + 64*a^2*b^6 - 40*b^8 )*cos(d*x + c)^6 + (a^8 - 22*a^6*b^2 - 11*a^4*b^4 + 152*a^2*b^6 - 120*b^8) *cos(d*x + c)^4 - (2*a^8 - 47*a^6*b^2 + 53*a^4*b^4 + 112*a^2*b^6 - 120*b^8 )*cos(d*x + c)^2 + 2*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7 + (a^7*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^4 - 2*(a^7*b - 25*a^5*b ^3 + 64*a^3*b^5 - 40*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^8 - 24*a^6*b^2 + 39*a^4*b^4 + 24*a^2*b^6 - 40*b^8 - (a^ 6*b^2 - 25*a^4*b^4 + 64*a^2*b^6 - 40*b^8)*cos(d*x + c)^6 + (a^8 - 22*a^6*b ^2 - 11*a^4*b^4 + 152*a^2*b^6 - 120*b^8)*cos(d*x + c)^4 - (2*a^8 - 47*a...
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(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 = 0.21 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.62 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/64*(24*(a^4 - 24*a^2*b^2 + 40*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 - 192*(2*a^4*b - 11*a^2*b^3 + 10*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*(7*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 + 6*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + a^2*b^4*tan(1/2*d*x + 1/2*c)^2 - 22*b^6*tan(1/2*d*x + 1/2*c)^2 + 17*a^3*b^3*tan(1/2*d*x + 1/2*c) - 32*a*b ^5*tan(1/2*d*x + 1/2*c) + 6*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) - (50*a^4*tan(1/2*d*x + 1/2*c)^4 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 2000*b^4*tan(1/2*d*x + 1/2*c)^4 + 120*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 - 8*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*ta n(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 - 8*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 + 120*a^8*b*tan(1/2*d*x + 1/2*c) - 32 0*a^6*b^3*tan(1/2*d*x + 1/2*c))/a^12)/d
Time = 34.42 (sec) , antiderivative size = 1487, normalized size of antiderivative = 4.37 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)*(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)^2*((3*(a^2 + 4*b^2)) /(32*a^5) + 1/(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) + 1/(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 - (tan(c/2 + (d*x) /2)^3*(19*a^4*b - 40*a^2*b^3) - tan(c/2 + (d*x)/2)^4*(448*a*b^4 + (15*a^5) /4 - 224*a^3*b^2) + tan(c/2 + (d*x)/2)^7*(30*a^4*b - 192*b^5 + 32*a^2*b^3) + tan(c/2 + (d*x)/2)^5*(50*a^4*b - 832*b^5 + 280*a^2*b^3) + a^5/4 - tan(c /2 + (d*x)/2)^2*((3*a^5)/2 - 5*a^3*b^2) - a^4*b*tan(c/2 + (d*x)/2) - (2*ta n(c/2 + (d*x)/2)^6*(a^6 + 176*b^6 + 152*a^2*b^4 - 114*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)) - (b*tan(c/2 + (d*x)/2)^3)/(8*a^4*d) + (log(tan(c/2 + (d *x)/2))*((3*a^4)/8 + 15*b^4 - 9*a^2*b^2))/(a^7*d) + (b*atan(((b*(-(a + b)* (a - b))^(1/2)*(((27*a^11*b)/4 + 60*a^7*b^5 - 51*a^9*b^3)/a^12 - (tan(c/2 + (d*x)/2)*(3*a^11 - 480*a^5*b^6 + 528*a^7*b^4 - 126*a^9*b^2))/(4*a^11) + (3*b*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^14 - 32 *a^12*b^2))/(4*a^11))*(2*a^4 + 10*b^4 - 11*a^2*b^2))/(2*(a^9 - a^7*b^2)))* (2*a^4 + 10*b^4 - 11*a^2*b^2)*3i)/(2*(a^9 - a^7*b^2)) - (b*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(3*a^11 - 480*a^5*b^6 + 528*a^7*b^4 - 126*a ^9*b^2))/(4*a^11) - ((27*a^11*b)/4 + 60*a^7*b^5 - 51*a^9*b^3)/a^12 + (3...
Time = 0.21 (sec) , antiderivative size = 1357, normalized size of antiderivative = 3.99 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+b*sin(d*x+c))^3,x)
Output:
( - 96*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(c + d*x)**6*a**4*b**3 + 528*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**5 - 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**7 - 1 92*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin( c + d*x)**5*a**5*b**2 + 1056*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**3*b**4 - 960*sqrt(a**2 - b**2)*at an((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a*b**6 - 96 *sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**6*b + 528*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqr t(a**2 - b**2))*sin(c + d*x)**4*a**4*b**3 - 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**5 - 104*c os(c + d*x)*sin(c + d*x)**5*a**5*b**3 + 344*cos(c + d*x)*sin(c + d*x)**5*a **3*b**5 - 240*cos(c + d*x)*sin(c + d*x)**5*a*b**7 - 166*cos(c + d*x)*sin( c + d*x)**4*a**6*b**2 + 526*cos(c + d*x)*sin(c + d*x)**4*a**4*b**4 - 360*c os(c + d*x)*sin(c + d*x)**4*a**2*b**6 - 44*cos(c + d*x)*sin(c + d*x)**3*a* *7*b + 124*cos(c + d*x)*sin(c + d*x)**3*a**5*b**3 - 80*cos(c + d*x)*sin(c + d*x)**3*a**3*b**5 + 10*cos(c + d*x)*sin(c + d*x)**2*a**8 - 30*cos(c + d* x)*sin(c + d*x)**2*a**6*b**2 + 20*cos(c + d*x)*sin(c + d*x)**2*a**4*b**4 + 8*cos(c + d*x)*sin(c + d*x)*a**7*b - 8*cos(c + d*x)*sin(c + d*x)*a**5*...