Integrand size = 29, antiderivative size = 329 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))} \] Output:
5*(a^2-4*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^6/d-5/2*b*(3*a^2-4*b^2)*arctanh(cos(d*x+c))/a^6/d+1/6*(3*a^4+35*a^2 *b^2-60*b^4)*cot(d*x+c)/a^5/b^2/d-cos(d*x+c)/b/d/(a+b*sin(d*x+c))^2-1/2*a* cot(d*x+c)/b^2/d/(a+b*sin(d*x+c))^2-1/3*(3*a^2-5*b^2)*cot(d*x+c)/a^3/d/(a+ b*sin(d*x+c))^2+5/6*b*cot(d*x+c)*csc(d*x+c)/a^2/d/(a+b*sin(d*x+c))^2-1/3*c ot(d*x+c)*csc(d*x+c)^2/a/d/(a+b*sin(d*x+c))^2-5/2*(a^2-2*b^2)*cot(d*x+c)/a ^4/d/(a+b*sin(d*x+c))
Time = 6.78 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}+\frac {\left (7 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-18 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^5 d}+\frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}-\frac {5 \left (3 a^2 b-4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {5 \left (3 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-7 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+18 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^5 d}+\frac {-a^4 \cos (c+d x)+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{2 a^4 b d (a+b \sin (c+d x))^2}+\frac {a^4 \cos (c+d x)+7 a^2 b^2 \cos (c+d x)-8 b^4 \cos (c+d x)}{2 a^5 b d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^3 d} \] Input:
Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^3,x]
Output:
(5*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) + ((7*a^2 *Cos[(c + d*x)/2] - 18*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^5*d) + (3*b*Csc[(c + d*x)/2]^2)/(8*a^4*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2 )/(24*a^3*d) - (5*(3*a^2*b - 4*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + (5* (3*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^6*d) - (3*b*Sec[(c + d*x)/2] ^2)/(8*a^4*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 18*b^2*Sin[(c + d*x)/2]))/(6*a^5*d) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b ^4*Cos[c + d*x])/(2*a^4*b*d*(a + b*Sin[c + d*x])^2) + (a^4*Cos[c + d*x] + 7*a^2*b^2*Cos[c + d*x] - 8*b^4*Cos[c + d*x])/(2*a^5*b*d*(a + b*Sin[c + d*x ])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^3*d)
Time = 2.76 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.26, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3375, 27, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^4 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int -\frac {2 \csc ^2(c+d x) \left (3 a^4+14 b^2 a^2+3 b \left (3 a^2-b^2\right ) \sin (c+d x) a-20 b^4-3 b^2 \left (2 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3}dx}{12 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\csc ^2(c+d x) \left (3 a^4+14 b^2 a^2+3 b \left (3 a^2-b^2\right ) \sin (c+d x) a-20 b^4-3 b^2 \left (2 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3}dx}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {3 a^4+14 b^2 a^2+3 b \left (3 a^2-b^2\right ) \sin (c+d x) a-20 b^4-3 b^2 \left (2 a^2-5 b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^2 (a+b \sin (c+d x))^3}dx}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int \frac {2 \csc ^2(c+d x) \left (3 a^6+17 b^2 a^4-50 b^4 a^2+b \left (6 a^4-11 b^2 a^2+5 b^4\right ) \sin (c+d x) a+30 b^6-4 b^2 \left (3 a^4-8 b^2 a^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^2(c+d x) \left (3 a^6+17 b^2 a^4-50 b^4 a^2+b \left (6 a^4-11 b^2 a^2+5 b^4\right ) \sin (c+d x) a+30 b^6-4 b^2 \left (3 a^4-8 b^2 a^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {3 a^6+17 b^2 a^4-50 b^4 a^2+b \left (6 a^4-11 b^2 a^2+5 b^4\right ) \sin (c+d x) a+30 b^6-4 b^2 \left (3 a^4-8 b^2 a^2+5 b^4\right ) \sin (c+d x)^2}{\sin (c+d x)^2 (a+b \sin (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\csc ^2(c+d x) \left (-15 b^2 \left (a^2-2 b^2\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+\left (3 a^4+35 b^2 a^2-60 b^4\right ) \left (a^2-b^2\right )^2+a b \left (3 a^2-10 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {-15 b^2 \left (a^2-2 b^2\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+\left (3 a^4+35 b^2 a^2-60 b^4\right ) \left (a^2-b^2\right )^2+a b \left (3 a^2-10 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int -\frac {15 \csc (c+d x) \left (\left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 b^3+a \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (a^2-b^2\right )^2 \left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \int \frac {\csc (c+d x) \left (\left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 b^3+a \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \int \frac {\left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 b^3+a \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^2}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \left (\frac {b^2 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}+\frac {b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}\right )}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \left (\frac {b^2 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}+\frac {b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}\right )}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \left (\frac {2 b^2 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}+\frac {b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}\right )}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \left (\frac {b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}-\frac {4 b^2 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}\right )}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {15 \left (\frac {b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}+\frac {2 b^2 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}}{6 a^2 b^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\frac {2 b^2 \left (3 a^2-5 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))^2}+\frac {\frac {15 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac {-\frac {15 \left (\frac {2 b^2 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}}{a \left (a^2-b^2\right )}}{6 a^2 b^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
Input:
Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^3,x]
Output:
-(Cos[c + d*x]/(b*d*(a + b*Sin[c + d*x])^2)) - (a*Cot[c + d*x])/(2*b^2*d*( a + b*Sin[c + d*x])^2) + (5*b*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*d*(a + b*S in[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x] )^2) - ((2*b^2*(3*a^2 - 5*b^2)*Cot[c + d*x])/(a*d*(a + b*Sin[c + d*x])^2) + (((-15*((2*b^2*(a^2 - 4*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^3*(3*a^2 - 4*b^2)*(a^2 - b^2)^ 2*ArcTanh[Cos[c + d*x]])/(a*d)))/a - ((a^2 - b^2)^2*(3*a^4 + 35*a^2*b^2 - 60*b^4)*Cot[c + d*x])/(a*d))/(a*(a^2 - b^2)) + (15*b^2*(a^2 - 2*b^2)*(a^2 - b^2)*Cot[c + d*x])/(a*d*(a + b*Sin[c + d*x])))/(a*(a^2 - b^2)))/(6*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 = 3.38 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-9 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5 b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (-\frac {a \left (a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}+\frac {9 b \left (a^{4}+a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \left (a^{4}+25 a^{2} b^{2}-26 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {9 a^{4} b}{2}-\frac {9 a^{2} b^{3}}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {5 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(359\) |
| default | \(\frac {\frac {\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-9 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5 b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (-\frac {a \left (a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}+\frac {9 b \left (a^{4}+a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \left (a^{4}+25 a^{2} b^{2}-26 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {9 a^{4} b}{2}-\frac {9 a^{2} b^{3}}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {5 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(359\) |
| risch | \(\frac {-60 i b^{6}+45 i a^{4} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-110 i b^{2} a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+20 i a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-210 i b^{2} a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+240 i b^{2} a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-45 i b^{4} a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+180 i b^{4} a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-190 i b^{4} a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-210 b^{5} a \,{\mathrm e}^{i \left (d x +c \right )}+3 i b^{2} a^{4}+35 i a^{2} b^{4}+15 a^{3} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+240 i b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+300 b^{5} a \,{\mathrm e}^{7 i \left (d x +c \right )}-60 i b^{6} {\mathrm e}^{8 i \left (d x +c \right )}+240 i b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-360 i b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-24 a^{5} b \,{\mathrm e}^{7 i \left (d x +c \right )}+36 a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}-24 b \,a^{5} {\mathrm e}^{3 i \left (d x +c \right )}+6 i a^{6} {\mathrm e}^{8 i \left (d x +c \right )}-18 i a^{6} {\mathrm e}^{6 i \left (d x +c \right )}+18 i a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 b \,a^{5} {\mathrm e}^{i \left (d x +c \right )}+125 b^{3} a^{3} {\mathrm e}^{i \left (d x +c \right )}-30 b^{5} a \,{\mathrm e}^{9 i \left (d x +c \right )}-150 b^{3} a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+6 a^{5} b \,{\mathrm e}^{9 i \left (d x +c \right )}-410 b^{3} a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+660 b^{5} a \,{\mathrm e}^{3 i \left (d x +c \right )}-720 b^{5} {\mathrm e}^{5 i \left (d x +c \right )} a +420 b^{3} a^{3} {\mathrm e}^{5 i \left (d x +c \right )}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} b^{2} d \,a^{5}}-\frac {15 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{6}}+\frac {15 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}-\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{6}}+\frac {5 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{2 d \,a^{4}}-\frac {10 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{6}}-\frac {5 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{2 d \,a^{4}}+\frac {10 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{6}}\) | \(878\) |
Input:
int(cos(d*x+c)^2*cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/8/a^5*(1/3*a^2*tan(1/2*d*x+1/2*c)^3-3*a*b*tan(1/2*d*x+1/2*c)^2-9*ta n(1/2*d*x+1/2*c)*a^2+24*b^2*tan(1/2*d*x+1/2*c))-1/24/a^3/tan(1/2*d*x+1/2*c )^3-1/8*(-9*a^2+24*b^2)/a^5/tan(1/2*d*x+1/2*c)+3/8/a^4*b/tan(1/2*d*x+1/2*c )^2+5/2/a^6*b*(3*a^2-4*b^2)*ln(tan(1/2*d*x+1/2*c))+2/a^6*((-1/2*a*(a^4-11* a^2*b^2+10*b^4)*tan(1/2*d*x+1/2*c)^3+9/2*b*(a^4+a^2*b^2-2*b^4)*tan(1/2*d*x +1/2*c)^2+1/2*a*(a^4+25*a^2*b^2-26*b^4)*tan(1/2*d*x+1/2*c)+9/2*a^4*b-9/2*a ^2*b^3)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+5/2*(a^4-5*a^2 *b^2+4*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. 735 vs. \(2 (310) = 620\).
Time = 0.35 (sec) , antiderivative size = 1553, normalized size of antiderivative = 4.72 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
[1/12*(2*(3*a^5 + 35*a^3*b^2 - 60*a*b^4)*cos(d*x + c)^5 - 20*(2*a^5 + 5*a^ 3*b^2 - 12*a*b^4)*cos(d*x + c)^3 - 15*(2*(a^3*b - 4*a*b^3)*cos(d*x + c)^4 + 2*a^3*b - 8*a*b^3 - 4*(a^3*b - 4*a*b^3)*cos(d*x + c)^2 + ((a^2*b^2 - 4*b ^4)*cos(d*x + c)^4 + a^4 - 3*a^2*b^2 - 4*b^4 - (a^4 - 2*a^2*b^2 - 8*b^4)*c os(d*x + c)^2)*sin(d*x + c))*sqrt(-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)) + 30*(a^5 + a^3*b^2 - 4*a*b^4)*cos(d*x + c) - 15*(6*a^3*b^2 - 8*a*b^4 + 2*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^4 - 4*(3*a^3*b^2 - 4*a*b ^4)*cos(d*x + c)^2 + (3*a^4*b - a^2*b^3 - 4*b^5 + (3*a^2*b^3 - 4*b^5)*cos( d*x + c)^4 - (3*a^4*b + 2*a^2*b^3 - 8*b^5)*cos(d*x + c)^2)*sin(d*x + c))*l og(1/2*cos(d*x + c) + 1/2) + 15*(6*a^3*b^2 - 8*a*b^4 + 2*(3*a^3*b^2 - 4*a* b^4)*cos(d*x + c)^4 - 4*(3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^2 + (3*a^4*b - a^2*b^3 - 4*b^5 + (3*a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - (3*a^4*b + 2*a^2*b^ 3 - 8*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 10 *((11*a^4*b - 18*a^2*b^3)*cos(d*x + c)^3 - 6*(2*a^4*b - 3*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/(2*a^7*b*d*cos(d*x + c)^4 - 4*a^7*b*d*cos(d*x + c)^2 + 2*a^7*b*d + (a^6*b^2*d*cos(d*x + c)^4 - (a^8 + 2*a^6*b^2)*d*cos(d*x + c) ^2 + (a^8 + a^6*b^2)*d)*sin(d*x + c)), 1/12*(2*(3*a^5 + 35*a^3*b^2 - 60*a* b^4)*cos(d*x + c)^5 - 20*(2*a^5 + 5*a^3*b^2 - 12*a*b^4)*cos(d*x + c)^3 ...
\[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cos(d*x+c)**2*cot(d*x+c)**4/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)**2*cot(c + d*x)**4/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
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 = 0.23 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/24*(60*(3*a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 + 120*(a^4 - 5*a^2*b^2 + 4*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*t an(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^6) - 24*(a^5 *tan(1/2*d*x + 1/2*c)^3 - 11*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + 10*a*b^4*tan (1/2*d*x + 1/2*c)^3 - 9*a^4*b*tan(1/2*d*x + 1/2*c)^2 - 9*a^2*b^3*tan(1/2*d *x + 1/2*c)^2 + 18*b^5*tan(1/2*d*x + 1/2*c)^2 - a^5*tan(1/2*d*x + 1/2*c) - 25*a^3*b^2*tan(1/2*d*x + 1/2*c) + 26*a*b^4*tan(1/2*d*x + 1/2*c) - 9*a^4*b + 9*a^2*b^3)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2 *a^6) + (a^6*tan(1/2*d*x + 1/2*c)^3 - 9*a^5*b*tan(1/2*d*x + 1/2*c)^2 - 27* a^6*tan(1/2*d*x + 1/2*c) + 72*a^4*b^2*tan(1/2*d*x + 1/2*c))/a^9 - (330*a^2 *b*tan(1/2*d*x + 1/2*c)^3 - 440*b^3*tan(1/2*d*x + 1/2*c)^3 - 27*a^3*tan(1/ 2*d*x + 1/2*c)^2 + 72*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 9*a^2*b*tan(1/2*d*x + 1/2*c) + a^3)/(a^6*tan(1/2*d*x + 1/2*c)^3))/d
Time = 22.09 (sec) , antiderivative size = 1082, normalized size of antiderivative = 3.29 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^2*cot(c + d*x)^4)/(a + b*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)^3/(24*a^3*d) + (tan(c/2 + (d*x)/2)^4*((77*a^4)/3 - 304* b^4 + 200*a^2*b^2) + tan(c/2 + (d*x)/2)^6*(a^4 - 80*b^4 + 64*a^2*b^2) - a^ 4/3 + tan(c/2 + (d*x)/2)^2*((25*a^4)/3 - (40*a^2*b^2)/3) - tan(c/2 + (d*x) /2)^3*(156*a*b^3 - (338*a^3*b)/3) - (3*tan(c/2 + (d*x)/2)^5*(48*b^5 - 37*a ^4*b + 8*a^2*b^3))/a + (5*a^3*b*tan(c/2 + (d*x)/2))/3)/(d*(8*a^7*tan(c/2 + (d*x)/2)^3 + 8*a^7*tan(c/2 + (d*x)/2)^7 + tan(c/2 + (d*x)/2)^5*(16*a^7 + 32*a^5*b^2) + 32*a^6*b*tan(c/2 + (d*x)/2)^4 + 32*a^6*b*tan(c/2 + (d*x)/2)^ 6)) - (tan(c/2 + (d*x)/2)*((3*(a^2 + 4*b^2))/(8*a^5) + 3/(4*a^3) - (9*b^2) /(2*a^5)))/d + (log(tan(c/2 + (d*x)/2))*(15*a^2*b - 20*b^3))/(2*a^6*d) - ( 3*b*tan(c/2 + (d*x)/2)^2)/(8*a^4*d) + (atan((((b^2 - a^2)^(1/2)*(a^2 - 4*b ^2)*((5*a^10 + 40*a^6*b^4 - 40*a^8*b^2)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8 *b + 80*a^4*b^5 - 100*a^6*b^3))/a^9 - (5*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6 *a^12 - 8*a^10*b^2))/a^9)*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2))/(2*a^6))*5i)/(2 *a^6) + ((b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((5*a^10 + 40*a^6*b^4 - 40*a^8*b^ 2)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8*b + 80*a^4*b^5 - 100*a^6*b^3))/a^9 + (5*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^12 - 8*a^10*b^2))/a^9)*(b^2 - a^2) ^(1/2)*(a^2 - 4*b^2))/(2*a^6))*5i)/(2*a^6))/((75*a^6*b - 400*b^7 + 800*a^2 *b^5 - 475*a^4*b^3)/a^10 + (2*tan(c/2 + (d*x)/2)*(25*a^6 - 200*b^6 + 350*a ^2*b^4 - 175*a^4*b^2))/a^9 + (5*(b^2 - a^2)^(1/2)*(a^2 - 4*b^2)*((5*a^10 + 40*a^6*b^4 - 40*a^8*b^2)/a^10 + (tan(c/2 + (d*x)/2)*(25*a^8*b + 80*a^4...
Time = 0.20 (sec) , antiderivative size = 824, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^2*cot(d*x+c)^4/(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))*si n(c + d*x)**5*a**2*b**3 - 960*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*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)**4*a**3*b**2 - 192 0*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a*b**4 + 240*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sq rt(a**2 - b**2))*sin(c + d*x)**3*a**4*b - 960*sqrt(a**2 - b**2)*atan((tan( (c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*a**2*b**3 + 24*cos( c + d*x)*sin(c + d*x)**4*a**5*b + 280*cos(c + d*x)*sin(c + d*x)**4*a**3*b* *3 - 480*cos(c + d*x)*sin(c + d*x)**4*a*b**5 + 440*cos(c + d*x)*sin(c + d* x)**3*a**4*b**2 - 720*cos(c + d*x)*sin(c + d*x)**3*a**2*b**4 + 112*cos(c + d*x)*sin(c + d*x)**2*a**5*b - 160*cos(c + d*x)*sin(c + d*x)**2*a**3*b**3 + 40*cos(c + d*x)*sin(c + d*x)*a**4*b**2 - 16*cos(c + d*x)*a**5*b + 360*lo g(tan((c + d*x)/2))*sin(c + d*x)**5*a**2*b**4 - 480*log(tan((c + d*x)/2))* sin(c + d*x)**5*b**6 + 720*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**3*b**3 - 960*log(tan((c + d*x)/2))*sin(c + d*x)**4*a*b**5 + 360*log(tan((c + d*x )/2))*sin(c + d*x)**3*a**4*b**2 - 480*log(tan((c + d*x)/2))*sin(c + d*x)** 3*a**2*b**4 + 25*sin(c + d*x)**5*a**4*b**2 + 120*sin(c + d*x)**5*a**2*b**4 - 240*sin(c + d*x)**5*b**6 + 50*sin(c + d*x)**4*a**5*b + 240*sin(c + d*x) **4*a**3*b**3 - 480*sin(c + d*x)**4*a*b**5 + 25*sin(c + d*x)**3*a**6 + ...