Integrand size = 31, antiderivative size = 245 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {55 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{512 d}-\frac {55 a \cot (c+d x)}{512 d \sqrt {a+a \sin (c+d x)}}-\frac {55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt {a+a \sin (c+d x)}}+\frac {329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt {a+a \sin (c+d x)}}+\frac {47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{6 d} \] Output:
-55/512*a^(1/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d-55/51 2*a*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-55/768*a*cot(d*x+c)*csc(d*x+c)/d/( a+a*sin(d*x+c))^(1/2)+329/960*a*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c)) ^(1/2)+47/160*a*cot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)-1/60*a*co t(d*x+c)*csc(d*x+c)^4/d/(a+a*sin(d*x+c))^(1/2)-1/6*cot(d*x+c)*csc(d*x+c)^5 *(a+a*sin(d*x+c))^(1/2)/d
Time = 9.82 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.98 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\csc ^{19}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-24540 \cos \left (\frac {1}{2} (c+d x)\right )-25684 \cos \left (\frac {3}{2} (c+d x)\right )-14490 \cos \left (\frac {5}{2} (c+d x)\right )-15006 \cos \left (\frac {7}{2} (c+d x)\right )-550 \cos \left (\frac {9}{2} (c+d x)\right )-1650 \cos \left (\frac {11}{2} (c+d x)\right )-8250 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12375 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-4950 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+825 \cos (6 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8250 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-12375 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4950 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-825 \cos (6 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24540 \sin \left (\frac {1}{2} (c+d x)\right )-25684 \sin \left (\frac {3}{2} (c+d x)\right )+14490 \sin \left (\frac {5}{2} (c+d x)\right )-15006 \sin \left (\frac {7}{2} (c+d x)\right )+550 \sin \left (\frac {9}{2} (c+d x)\right )-1650 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{7680 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]
Output:
(Csc[(c + d*x)/2]^19*Sqrt[a*(1 + Sin[c + d*x])]*(-24540*Cos[(c + d*x)/2] - 25684*Cos[(3*(c + d*x))/2] - 14490*Cos[(5*(c + d*x))/2] - 15006*Cos[(7*(c + d*x))/2] - 550*Cos[(9*(c + d*x))/2] - 1650*Cos[(11*(c + d*x))/2] - 8250 *Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12375*Cos[2*(c + d*x)]*Log [1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4950*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 825*Cos[6*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8250*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 12375*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x) /2]] + 4950*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 825*Cos[6*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2454 0*Sin[(c + d*x)/2] - 25684*Sin[(3*(c + d*x))/2] + 14490*Sin[(5*(c + d*x))/ 2] - 15006*Sin[(7*(c + d*x))/2] + 550*Sin[(9*(c + d*x))/2] - 1650*Sin[(11* (c + d*x))/2]))/(7680*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[( c + d*x)/4]^2)^6)
Time = 2.49 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.53, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.774, Rules used = {3042, 3360, 3042, 3251, 3042, 3251, 3042, 3252, 219, 3523, 27, 3042, 3459, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 \sqrt {a \sin (c+d x)+a}}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \csc ^7(c+d x) \sqrt {\sin (c+d x) a+a} \left (1-2 \sin ^2(c+d x)\right )dx+\int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx+\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx+\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^7}dx+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {1}{2} \csc ^6(c+d x) (a-15 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{6 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc ^6(c+d x) (a-15 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-15 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^6}dx}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {-\frac {141}{10} a \int \csc ^5(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {141}{10} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^5}dx-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {a^2 \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {141}{10} a \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{12 a}+\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{6 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]
Output:
-1/2*(a*Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(6*d) + (3*(-((Sqrt[a]*ArcT anh[(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 + (-1/5*(a^2*Cot[c + d*x]*Csc[c + d*x]^ 4)/(d*Sqrt[a + a*Sin[c + d*x]]) - (141*a*(-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*a)
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 = 0.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (825 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {5}{2}}-4675 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {7}{2}}-825 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{8} \sin \left (d x +c \right )^{6}+7818 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {9}{2}}-1398 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {11}{2}}-4675 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {13}{2}}+825 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {15}{2}}\right )}{7680 a^{\frac {15}{2}} \sin \left (d x +c \right )^{6} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(198\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBO SE)
Output:
1/7680*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(15/2)*(825*(-a*(sin(d*x +c)-1))^(11/2)*a^(5/2)-4675*(-a*(sin(d*x+c)-1))^(9/2)*a^(7/2)-825*arctanh( (-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^8*sin(d*x+c)^6+7818*(-a*(sin(d*x+c)-1 ))^(7/2)*a^(9/2)-1398*(-a*(sin(d*x+c)-1))^(5/2)*a^(11/2)-4675*(-a*(sin(d*x +c)-1))^(3/2)*a^(13/2)+825*(-a*(sin(d*x+c)-1))^(1/2)*a^(15/2))/sin(d*x+c)^ 6/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. 525 vs. \(2 (213) = 426\).
Time = 0.10 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.14 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {825 \, {\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} + {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \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 (825 \, \cos \left (d x + c\right )^{6} + 550 \, \cos \left (d x + c\right )^{5} + 707 \, \cos \left (d x + c\right )^{4} + 1156 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + {\left (825 \, \cos \left (d x + c\right )^{5} + 275 \, \cos \left (d x + c\right )^{4} + 982 \, \cos \left (d x + c\right )^{3} - 174 \, \cos \left (d x + c\right )^{2} - 399 \, \cos \left (d x + c\right ) + 27\right )} \sin \left (d x + c\right ) - 426 \, \cos \left (d x + c\right ) - 27\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{30720 \, {\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 ) + {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="f ricas")
Output:
1/30720*(825*(cos(d*x + c)^7 + cos(d*x + c)^6 - 3*cos(d*x + c)^5 - 3*cos(d *x + c)^4 + 3*cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^6 - 3*cos( d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(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)*sq rt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(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*(825*cos(d*x + c)^6 + 550*cos(d*x + c)^5 + 707*cos(d*x + c)^4 + 1156*cos(d*x + c)^3 - 225*cos(d*x + c)^2 + (825*cos (d*x + c)^5 + 275*cos(d*x + c)^4 + 982*cos(d*x + c)^3 - 174*cos(d*x + c)^2 - 399*cos(d*x + c) + 27)*sin(d*x + c) - 426*cos(d*x + c) - 27)*sqrt(a*sin (d*x + c) + a))/(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*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*s in(d*x + c) - d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**3*(a+a*sin(d*x+c))**(1/2),x)
Output:
Timed out
\[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="m axima")
Output:
integrate(sqrt(a*sin(d*x + c) + a)*cot(d*x + c)^4*csc(d*x + c)^3, x)
Time = 0.17 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.11 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (825 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (26400 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 74800 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 62544 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5592 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9350 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 825 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}\right )} \sqrt {a}}{30720 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="g iac")
Output:
-1/30720*sqrt(2)*(825*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*(26400*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin (-1/4*pi + 1/2*d*x + 1/2*c)^11 - 74800*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) *sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 + 62544*sgn(cos(-1/4*pi + 1/2*d*x + 1/2* c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 5592*sgn(cos(-1/4*pi + 1/2*d*x + 1/ 2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 9350*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 825*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)^6)*sqrt(a)/d
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^7} \,d x \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^3,x)
Output:
int(((sin(c + d*x)^2 - 1)^2*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^7, x)
\[ \int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{3}d x \right ) \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4*csc(c + d*x)**3,x)