Integrand size = 31, antiderivative size = 281 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {61 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d} \]
-61/1024*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))*a^(1/2)/d-61/1 024*a*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-61/1536*a*cot(d*x+c)*csc(d*x+c)/ d/(a+a*sin(d*x+c))^(1/2)-61/1920*a*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+ c))^(1/2)+579/2240*a*cot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)+193/ 840*a*cot(d*x+c)*csc(d*x+c)^4/d/(a+a*sin(d*x+c))^(1/2)-1/84*a*cot(d*x+c)*c sc(d*x+c)^5/d/(a+a*sin(d*x+c))^(1/2)-1/7*cot(d*x+c)*csc(d*x+c)^6*(a+a*sin( d*x+c))^(1/2)/d
Time = 1.96 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.68 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left (-102480 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+102480 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^7(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (-201298-244533 \cos (2 (c+d x))-52094 \cos (4 (c+d x))+6405 \cos (6 (c+d x))+49128 \sin (c+d x)-179636 \sin (3 (c+d x))-8540 \sin (5 (c+d x)))\right )}{3440640 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
(Sqrt[a*(1 + Sin[c + d*x])]*(-102480*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d *x)/2]] + 102480*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Csc[c + d* x]^7*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(-201298 - 244533*Cos[2*(c + d* x)] - 52094*Cos[4*(c + d*x)] + 6405*Cos[6*(c + d*x)] + 49128*Sin[c + d*x] - 179636*Sin[3*(c + d*x)] - 8540*Sin[5*(c + d*x)])))/(3440640*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 2.91 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.62, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.903, Rules used = {3042, 3360, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3252, 219, 3523, 27, 3042, 3459, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3252, 219}
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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 \sqrt {a \sin (c+d x)+a}}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \csc ^8(c+d x) \sqrt {\sin (c+d x) a+a} \left (1-2 \sin ^2(c+d x)\right )dx+\int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx+\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx+\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx+\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx+\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx+\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^8}dx+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {1}{2} \csc ^7(c+d x) (a-17 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{7 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc ^7(c+d x) (a-17 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-17 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^7}dx}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {-\frac {193}{12} a \int \csc ^6(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {193}{12} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^6}dx-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \int \csc ^5(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^5}dx-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {193}{12} a \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}+\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\) |
-1/3*(a*Cot[c + d*x]*Csc[c + d*x]^2)/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(7*d) + (5*(-1/2*(a*Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (3*(-((Sqrt[a]*ArcTan h[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/ (d*Sqrt[a + a*Sin[c + d*x]])))/4))/6 + (-1/6*(a^2*Cot[c + d*x]*Csc[c + d*x ]^5)/(d*Sqrt[a + a*Sin[c + d*x]]) - (193*a*(-1/5*(a*Cot[c + d*x]*Csc[c + d *x]^4)/(d*Sqrt[a + a*Sin[c + d*x]]) + (9*(-1/4*(a*Cot[c + d*x]*Csc[c + d*x ]^3)/(d*Sqrt[a + a*Sin[c + d*x]]) + (7*(-1/3*(a*Cot[c + d*x]*Csc[c + d*x]^ 2)/(d*Sqrt[a + a*Sin[c + d*x]]) + (5*(-1/2*(a*Cot[c + d*x]*Csc[c + d*x])/( d*Sqrt[a + a*Sin[c + d*x]]) + (3*(-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x] )/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d *x]])))/4))/6))/8))/10))/12)/(14*a)
3.5.52.3.1 Defintions of rubi rules used
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/d^4 Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IGtQ[m, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (6405 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {7}{2}}-42700 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {9}{2}}+120841 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {11}{2}}+6405 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{10} \left (\sin ^{7}\left (d x +c \right )\right )-156672 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {13}{2}}+51191 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {15}{2}}+42700 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {17}{2}}-6405 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {19}{2}}\right )}{107520 a^{\frac {19}{2}} \sin \left (d x +c \right )^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(216\) |
-1/107520*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(19/2)*(6405*(-a*(sin (d*x+c)-1))^(13/2)*a^(7/2)-42700*(-a*(sin(d*x+c)-1))^(11/2)*a^(9/2)+120841 *(-a*(sin(d*x+c)-1))^(9/2)*a^(11/2)+6405*arctanh((-a*(sin(d*x+c)-1))^(1/2) /a^(1/2))*a^10*sin(d*x+c)^7-156672*(-a*(sin(d*x+c)-1))^(7/2)*a^(13/2)+5119 1*(-a*(sin(d*x+c)-1))^(5/2)*a^(15/2)+42700*(-a*(sin(d*x+c)-1))^(3/2)*a^(17 /2)-6405*(-a*(sin(d*x+c)-1))^(1/2)*a^(19/2))/sin(d*x+c)^7/cos(d*x+c)/(a+a* sin(d*x+c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (245) = 490\).
Time = 0.29 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.02 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {6405 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{7} + \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (6405 \, \cos \left (d x + c\right )^{7} + 2135 \, \cos \left (d x + c\right )^{6} - 22631 \, \cos \left (d x + c\right )^{5} - 37613 \, \cos \left (d x + c\right )^{4} + 1343 \, \cos \left (d x + c\right )^{3} + 27477 \, \cos \left (d x + c\right )^{2} - {\left (6405 \, \cos \left (d x + c\right )^{6} + 4270 \, \cos \left (d x + c\right )^{5} - 18361 \, \cos \left (d x + c\right )^{4} + 19252 \, \cos \left (d x + c\right )^{3} + 20595 \, \cos \left (d x + c\right )^{2} - 6882 \, \cos \left (d x + c\right ) - 7359\right )} \sin \left (d x + c\right ) - 477 \, \cos \left (d x + c\right ) - 7359\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{430080 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]
1/430080*(6405*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*c os(d*x + c)^2 - (cos(d*x + c)^7 + cos(d*x + c)^6 - 3*cos(d*x + c)^5 - 3*co s(d*x + c)^4 + 3*cos(d*x + c)^3 + 3*cos(d*x + c)^2 - cos(d*x + c) - 1)*sin (d*x + c) + 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos (d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a *sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*co s(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos( d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(6405*cos(d*x + c)^7 + 2135*cos(d*x + c)^6 - 22631*cos(d*x + c)^5 - 37613*cos(d*x + c)^4 + 134 3*cos(d*x + c)^3 + 27477*cos(d*x + c)^2 - (6405*cos(d*x + c)^6 + 4270*cos( d*x + c)^5 - 18361*cos(d*x + c)^4 + 19252*cos(d*x + c)^3 + 20595*cos(d*x + c)^2 - 6882*cos(d*x + c) - 7359)*sin(d*x + c) - 477*cos(d*x + c) - 7359)* sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos (d*x + c)^4 - 4*d*cos(d*x + c)^2 - (d*cos(d*x + c)^7 + d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 + 3*d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.07 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (6405 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (409920 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1366400 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1933456 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1253376 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 204764 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 85400 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6405 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}\right )} \sqrt {a}}{430080 \, d} \]
-1/430080*sqrt(2)*(6405*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d *x + 1/2*c))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(-1 /4*pi + 1/2*d*x + 1/2*c)) + 4*(409920*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))* sin(-1/4*pi + 1/2*d*x + 1/2*c)^13 - 1366400*sgn(cos(-1/4*pi + 1/2*d*x + 1/ 2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^11 + 1933456*sgn(cos(-1/4*pi + 1/2*d* x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 1253376*sgn(cos(-1/4*pi + 1 /2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 + 204764*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 85400*sgn(cos(-1/4 *pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 6405*sgn(cos(-1 /4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c))/(2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^7)*sqrt(a)/d
Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^8} \,d x \]