Integrand size = 21, antiderivative size = 289 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {b \left (9 a^2-20 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))} \] Output:
(2*a^4-19*a^2*b^2+20*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2)) /a^6/(a^2-b^2)^(1/2)/d-1/2*b*(9*a^2-20*b^2)*arctanh(cos(d*x+c))/a^6/d+1/6* (17*a^2-60*b^2)*cot(d*x+c)/a^5/d-(a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)/a^4/b/d +1/6*(3*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^2-1/3*co t(d*x+c)*csc(d*x+c)^2/a/d/(a+b*sin(d*x+c))^2+1/6*(3*a^2-20*b^2)*cot(d*x+c) *csc(d*x+c)/a^3/b/d/(a+b*sin(d*x+c))
Time = 6.20 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.59 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}+\frac {\left (2 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d}+\frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {\left (-9 a^2 b+20 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (9 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-2 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac {3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^3 d} \] Input:
Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]
Output:
((2*a^4 - 19*a^2*b^2 + 20*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) + ((2*a ^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(3*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) + ((-9*a^2*b + 20*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + (( 9*a^2*b - 20*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]*(-2*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(3*a^5*d) + (a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])/(2*a^4*d*( a + b*Sin[c + d*x])^2) + (3*a^2*b*Cos[c + d*x] - 8*b^3*Cos[c + d*x])/(2*a^ 5*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^3* d)
Time = 2.32 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.24, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 3203, 3042, 3534, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3203 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (-3 \left (a^2-5 b^2\right ) \sin ^2(c+d x)-2 a b \sin (c+d x)+2 \left (3 a^2-10 b^2\right )\right )}{(a+b \sin (c+d x))^2}dx}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-3 \left (a^2-5 b^2\right ) \sin (c+d x)^2-2 a b \sin (c+d x)+2 \left (3 a^2-10 b^2\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^3(c+d x) \left (-2 \left (3 a^4-23 b^2 a^2+20 b^4\right ) \sin ^2(c+d x)-5 a b \left (a^2-b^2\right ) \sin (c+d x)+12 \left (a^4-6 b^2 a^2+5 b^4\right )\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-2 \left (3 a^4-23 b^2 a^2+20 b^4\right ) \sin (c+d x)^2-5 a b \left (a^2-b^2\right ) \sin (c+d x)+12 \left (a^4-6 b^2 a^2+5 b^4\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {2 \csc ^2(c+d x) \left (-10 a \left (a^2-b^2\right ) \sin (c+d x) b^2-6 \left (a^4-6 b^2 a^2+5 b^4\right ) \sin ^2(c+d x) b+\left (17 a^4-77 b^2 a^2+60 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-10 a \left (a^2-b^2\right ) \sin (c+d x) b^2-6 \left (a^4-6 b^2 a^2+5 b^4\right ) \sin ^2(c+d x) b+\left (17 a^4-77 b^2 a^2+60 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-10 a \left (a^2-b^2\right ) \sin (c+d x) b^2-6 \left (a^4-6 b^2 a^2+5 b^4\right ) \sin (c+d x)^2 b+\left (17 a^4-77 b^2 a^2+60 b^4\right ) b}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int -\frac {3 \csc (c+d x) \left (\left (9 a^4-29 b^2 a^2+20 b^4\right ) b^2+2 a \left (a^4-6 b^2 a^2+5 b^4\right ) \sin (c+d x) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \int \frac {\csc (c+d x) \left (\left (9 a^4-29 b^2 a^2+20 b^4\right ) b^2+2 a \left (a^4-6 b^2 a^2+5 b^4\right ) \sin (c+d x) b\right )}{a+b \sin (c+d x)}dx}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \int \frac {\left (9 a^4-29 b^2 a^2+20 b^4\right ) b^2+2 a \left (a^4-6 b^2 a^2+5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (\frac {b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right ) \int \csc (c+d x)dx}{a}+\frac {b \left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (\frac {b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right ) \int \csc (c+d x)dx}{a}+\frac {b \left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (\frac {b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right ) \int \csc (c+d x)dx}{a}+\frac {2 b \left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (\frac {b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {4 b \left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}\right )}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (\frac {b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right ) \int \csc (c+d x)dx}{a}+\frac {2 b \left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}\right )}{a}-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{6 a^2 b}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}+\frac {\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}+\frac {-\frac {6 \left (a^4-6 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}-\frac {-\frac {b \left (17 a^4-77 a^2 b^2+60 b^4\right ) \cot (c+d x)}{a d}-\frac {3 \left (\frac {2 b \left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{6 a^2 b}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}\) |
Input:
Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]
Output:
((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x] )^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((-( ((-3*((2*b*(2*a^6 - 21*a^4*b^2 + 39*a^2*b^4 - 20*b^6)*ArcTan[(2*b + 2*a*Ta n[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (b^2*(9*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (b*(17*a^4 - 77* a^2*b^2 + 60*b^4)*Cot[c + d*x])/(a*d))/a) - (6*(a^4 - 6*a^2*b^2 + 5*b^4)*C ot[c + d*x]*Csc[c + d*x])/(a*d))/(a*(a^2 - b^2)) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a*d*(a + b*Sin[c + d*x])))/(6*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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[(3*a^2 + b^2*(m - 2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b*f*(m + 1)*Sin[e + f*x]^2)), x] - Simp[1/(3*a^2*b*( m + 1)) Int[((a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3)*Simp[6*a^2 - b^ 2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[ e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && L tQ[m, -1] && IntegerQ[2*m]
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 = 3.21 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (9 a^{2}-20 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {b \left (4 a^{4}-a^{2} b^{2}-18 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \,b^{2} \left (11 a^{2}-26 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{2}-9 b^{2}\right )}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{4}-19 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(360\) |
| default | \(\frac {\frac {\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (9 a^{2}-20 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {b \left (4 a^{4}-a^{2} b^{2}-18 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {a \,b^{2} \left (11 a^{2}-26 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{2}-9 b^{2}\right )}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{4}-19 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(360\) |
| risch | \(\frac {252 i b^{2} a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+240 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-298 i b^{2} a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-30 a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+92 i b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+18 i a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-63 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )} b^{2}-60 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+300 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}+240 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-60 i b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+17 i a^{2} b^{2}+204 a^{3} {\mathrm e}^{5 i \left (d x +c \right )} b -720 b^{3} a \,{\mathrm e}^{5 i \left (d x +c \right )}+102 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-60 i b^{4}-50 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-212 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+660 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-102 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-360 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+62 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-210 b^{3} a \,{\mathrm e}^{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} d \,a^{5}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,a^{2}}+\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,a^{4}}-\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,a^{6}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,a^{2}}-\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,a^{4}}+\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,a^{6}}+\frac {9 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 {9 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}}\) | \(924\) |
Input:
int(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)^2-5*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*(-5*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+1/2/a^6*b*(9*a^2-20*b^2)*ln(tan(1/2*d*x+1/2*c))+2/a^6*(((5/2*a^3*b^2-5 *a*b^4)*tan(1/2*d*x+1/2*c)^3+1/2*b*(4*a^4-a^2*b^2-18*b^4)*tan(1/2*d*x+1/2* c)^2+1/2*a*b^2*(11*a^2-26*b^2)*tan(1/2*d*x+1/2*c)+1/2*a^2*b*(4*a^2-9*b^2)) /(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(2*a^4-19*a^2*b^2 +20*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. 972 vs. \(2 (274) = 548\).
Time = 0.43 (sec) , antiderivative size = 2027, normalized size of antiderivative = 7.01 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
[1/12*(2*(17*a^5*b^2 - 77*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^5 - 4*(4*a^7 + 3*a^5*b^2 - 67*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^3 - 3*(4*a^5*b - 38*a^3*b^ 3 + 40*a*b^5 + 2*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^4 - 4*(2*a ^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 17*a^4*b^2 + a^2*b ^4 + 20*b^6 + (2*a^4*b^2 - 19*a^2*b^4 + 20*b^6)*cos(d*x + c)^4 - (2*a^6 - 15*a^4*b^2 - 18*a^2*b^4 + 40*b^6)*cos(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)) + 6*(2*a^7 - 3*a^5*b^2 - 19*a^3*b^4 + 20*a*b^6)*cos(d*x + c) - 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b ^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*co s(d*x + c) + 1/2) + 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a* b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4* b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2 *b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*(14*a^6*b - 59*a^4*b^3 + 45*a^2*b^5)*cos(d*x + c)^3 - 3*(11*a^6*...
\[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cot(d*x+c)**4/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**4/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^4/(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 = 0.20 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {12 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {24 \, {\left (2 \, a^{4} - 19 \, a^{2} b^{2} + 20 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} + \frac {24 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 26 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b - 9 \, a^{2} b^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a^{6}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/24*(12*(9*a^2*b - 20*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 + 24*(2*a^4 - 19*a^2*b^2 + 20*b^4)*(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^6) + 24* (5*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 + 4*a^ 4*b*tan(1/2*d*x + 1/2*c)^2 - 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 + 11*a^3*b^2*tan(1/2*d*x + 1/2*c) - 26*a*b^4*tan(1/2*d*x + 1/2*c) + 4*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 - 15*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 - (198*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 440*b^3*tan(1/2*d*x + 1/2*c )^3 - 15*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 = 34.15 (sec) , antiderivative size = 1261, normalized size of antiderivative = 4.36 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(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)*((3*(a^2 + 4*b^2))/( 8*a^5) + 1/(4*a^3) - (9*b^2)/(2*a^5)))/d + (tan(c/2 + (d*x)/2)^6*(5*a^4 - 80*b^4 + 16*a^2*b^2) + tan(c/2 + (d*x)/2)^4*((29*a^4)/3 - 304*b^4 + 72*a^2 *b^2) - a^4/3 + tan(c/2 + (d*x)/2)^2*((13*a^4)/3 - (40*a^2*b^2)/3) - tan(c /2 + (d*x)/2)^3*(156*a*b^3 - (170*a^3*b)/3) - (tan(c/2 + (d*x)/2)^5*(144*b ^5 - 55*a^4*b + 104*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)) + (log(tan(c/2 + (d*x)/2))*(9*a^2*b - 20*b^3))/(2*a^6*d) - (3*b*tan(c/2 + (d*x)/2)^2)/(8*a^4*d) + (atan((((-(a + b)*(a - b))^(1/2)*(a ^4 + 10*b^4 - (19*a^2*b^2)/2)*((2*a^10 + 40*a^6*b^4 - 28*a^8*b^2)/a^10 + ( tan(c/2 + (d*x)/2)*(13*a^8*b + 80*a^4*b^5 - 76*a^6*b^3))/a^9 + ((-(a + b)* (a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^12 - 8*a^10*b^2))/a^9)* (a^4 + 10*b^4 - (19*a^2*b^2)/2))/(a^8 - a^6*b^2))*1i)/(a^8 - a^6*b^2) + (( -(a + b)*(a - b))^(1/2)*(a^4 + 10*b^4 - (19*a^2*b^2)/2)*((2*a^10 + 40*a^6* b^4 - 28*a^8*b^2)/a^10 + (tan(c/2 + (d*x)/2)*(13*a^8*b + 80*a^4*b^5 - 76*a ^6*b^3))/a^9 - ((-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6 *a^12 - 8*a^10*b^2))/a^9)*(a^4 + 10*b^4 - (19*a^2*b^2)/2))/(a^8 - a^6*b^2) )*1i)/(a^8 - a^6*b^2))/((18*a^6*b - 400*b^7 + 560*a^2*b^5 - 211*a^4*b^3)/a ^10 + (2*tan(c/2 + (d*x)/2)*(4*a^6 - 200*b^6 + 230*a^2*b^4 - 58*a^4*b^2...
Time = 0.22 (sec) , antiderivative size = 1227, normalized size of antiderivative = 4.25 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^4/(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)**5*a**4*b**3 - 912*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**2*b**5 + 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*b**7 + 192* sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**5*b**2 - 1824*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**4 + 1920*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**6 + 96*s qrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*a**6*b - 912*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt( a**2 - b**2))*sin(c + d*x)**3*a**4*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)**3*a**2*b**5 + 136*cos (c + d*x)*sin(c + d*x)**4*a**5*b**3 - 616*cos(c + d*x)*sin(c + d*x)**4*a** 3*b**5 + 480*cos(c + d*x)*sin(c + d*x)**4*a*b**7 + 224*cos(c + d*x)*sin(c + d*x)**3*a**6*b**2 - 944*cos(c + d*x)*sin(c + d*x)**3*a**4*b**4 + 720*cos (c + d*x)*sin(c + d*x)**3*a**2*b**6 + 64*cos(c + d*x)*sin(c + d*x)**2*a**7 *b - 224*cos(c + d*x)*sin(c + d*x)**2*a**5*b**3 + 160*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 + 40*cos(c + d*x)*sin(c + d*x)*a**6*b**2 - 40*cos(c + d *x)*sin(c + d*x)*a**4*b**4 - 16*cos(c + d*x)*a**7*b + 16*cos(c + d*x)*a**5 *b**3 + 216*log(tan((c + d*x)/2))*sin(c + d*x)**5*a**4*b**4 - 696*log(t...