Integrand size = 29, antiderivative size = 170 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {11 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 \sqrt {a} d}-\frac {11 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}} \] Output:
-11/64*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(1/2)/d-11/64* cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+53/96*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin (d*x+c))^(1/2)+1/24*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c))^(1/2)-1/4*c ot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(374\) vs. \(2(170)=340\).
Time = 2.35 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.20 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\csc ^{12}\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (214 \cos \left (\frac {1}{2} (c+d x)\right )-558 \cos \left (\frac {3}{2} (c+d x)\right )-490 \cos \left (\frac {5}{2} (c+d x)\right )+66 \cos \left (\frac {7}{2} (c+d x)\right )-99 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+132 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-33 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+99 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-132 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+33 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-214 \sin \left (\frac {1}{2} (c+d x)\right )-558 \sin \left (\frac {3}{2} (c+d x)\right )+490 \sin \left (\frac {5}{2} (c+d x)\right )+66 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{192 d \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4 \sqrt {a (1+\sin (c+d x))}} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]
Output:
(Csc[(c + d*x)/2]^12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(214*Cos[(c + d *x)/2] - 558*Cos[(3*(c + d*x))/2] - 490*Cos[(5*(c + d*x))/2] + 66*Cos[(7*( c + d*x))/2] - 99*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 132*Cos[2 *(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 33*Cos[4*(c + d *x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 99*Log[1 - Cos[(c + d* x)/2] + Sin[(c + d*x)/2]] - 132*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 33*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[( c + d*x)/2]] - 214*Sin[(c + d*x)/2] - 558*Sin[(3*(c + d*x))/2] + 490*Sin[( 5*(c + d*x))/2] + 66*Sin[(7*(c + d*x))/2]))/(192*d*(Csc[(c + d*x)/4]^2 - S ec[(c + d*x)/4]^2)^4*Sqrt[a*(1 + Sin[c + d*x])])
Time = 2.45 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.97, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3360, 3042, 3259, 3042, 3128, 219, 3252, 219, 3523, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3464, 3042, 3128, 219, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a \sin (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^5 \sqrt {a \sin (c+d x)+a}}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \frac {\csc ^5(c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {\csc (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3259 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx-\int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {\int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {2 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle -\frac {2 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int -\frac {\csc ^4(c+d x) (9 \sin (c+d x) a+a)}{2 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\csc ^4(c+d x) (9 \sin (c+d x) a+a)}{\sqrt {\sin (c+d x) a+a}}dx}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {9 \sin (c+d x) a+a}{\sin (c+d x)^4 \sqrt {\sin (c+d x) a+a}}dx}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^3(c+d x) \left (5 \sin (c+d x) a^2+53 a^2\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^3(c+d x) \left (5 \sin (c+d x) a^2+53 a^2\right )}{\sqrt {\sin (c+d x) a+a}}dx}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {5 \sin (c+d x) a^2+53 a^2}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {\frac {\int -\frac {3 \csc ^2(c+d x) \left (11 a^3-53 a^3 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {3 \int \frac {\csc ^2(c+d x) \left (11 a^3-53 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {3 \int \frac {11 a^3-53 a^3 \sin (c+d x)}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (\frac {\int -\frac {\csc (c+d x) \left (117 a^4-11 a^4 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {\int \frac {\csc (c+d x) \left (117 a^4-11 a^4 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {\int \frac {117 a^4-11 a^4 \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {117 a^3 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-128 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {117 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-128 a^4 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {117 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {256 a^4 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {117 a^3 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {128 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (-\frac {\frac {128 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {234 a^4 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {-\frac {53 a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}-\frac {3 \left (-\frac {\frac {128 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {234 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {11 a^3 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{4 a}}{6 a}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]
Output:
(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x] ])])/(Sqrt[a]*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*d*Sqrt[a + a*Sin[c + d *x]]) - (-1/3*(a*Cot[c + d*x]*Csc[c + d*x]^2)/(d*Sqrt[a + a*Sin[c + d*x]]) + ((-53*a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]]) - ( 3*(-1/2*((-234*a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d *x]]])/d + (128*Sqrt[2]*a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sq rt[a + a*Sin[c + d*x]])])/d)/a - (11*a^3*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])))/(4*a))/(6*a))/(8*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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
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[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] - Simp[d/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*S in[e + f*x]), 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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(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/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), 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]
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 = 0.34 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (33 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {3}{2}}-33 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{5} \sin \left (d x +c \right )^{4}+7 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {5}{2}}-121 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {7}{2}}+33 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {9}{2}}\right )}{192 a^{\frac {11}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(162\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE )
Output:
1/192*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(11/2)*(33*(-a*(sin(d*x+c )-1))^(7/2)*a^(3/2)-33*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^5*sin( d*x+c)^4+7*(-a*(sin(d*x+c)-1))^(5/2)*a^(5/2)-121*(-a*(sin(d*x+c)-1))^(3/2) *a^(7/2)+33*(-a*(sin(d*x+c)-1))^(1/2)*a^(9/2))/sin(d*x+c)^4/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. 426 vs. \(2 (146) = 292\).
Time = 0.10 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.51 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {33 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (33 \, \cos \left (d x + c\right )^{4} - 106 \, \cos \left (d x + c\right )^{3} - 164 \, \cos \left (d x + c\right )^{2} + {\left (33 \, \cos \left (d x + c\right )^{3} + 139 \, \cos \left (d x + c\right )^{2} - 25 \, \cos \left (d x + c\right ) - 83\right )} \sin \left (d x + c\right ) + 58 \, \cos \left (d x + c\right ) + 83\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{768 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d + {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(1/2),x, algorithm="fri cas")
Output:
1/768*(33*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*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)*sqrt(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*(33*cos(d*x + c)^4 - 106*co s(d*x + c)^3 - 164*cos(d*x + c)^2 + (33*cos(d*x + c)^3 + 139*cos(d*x + c)^ 2 - 25*cos(d*x + c) - 83)*sin(d*x + c) + 58*cos(d*x + c) + 83)*sqrt(a*sin( d*x + c) + a))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d + (a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c))
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)/(a+a*sin(d*x+c))**(1/2),x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)/sqrt(a*(sin(c + d*x) + 1)), x)
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4} \csc \left (d x + c\right )}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(1/2),x, algorithm="max ima")
Output:
integrate(cot(d*x + c)^4*csc(d*x + c)/sqrt(a*sin(d*x + c) + a), x)
Time = 0.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\frac {33 \, \log \left (\frac {{\left | -128 \, \sqrt {2} - 256 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 128 \, \sqrt {2} - 256 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (264 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 28 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 242 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{384 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(1/2),x, algorithm="gia c")
Output:
1/384*(33*log(abs(-128*sqrt(2) - 256*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(1 28*sqrt(2) - 256*sin(-1/4*pi + 1/2*d*x + 1/2*c)))/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 2*sqrt(2)*(264*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2 *c)^7 + 28*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 242*sqrt(a)*sin(-1/4 *pi + 1/2*d*x + 1/2*c)^3 + 33*sqrt(a)*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)^4*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2* c))))/d
Timed out. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2}{{\sin \left (c+d\,x\right )}^5\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)*(a + a*sin(c + d*x))^(1/2)),x)
Output:
int((sin(c + d*x)^2 - 1)^2/(sin(c + d*x)^5*(a + a*sin(c + d*x))^(1/2)), x)
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )}{\sin \left (d x +c \right )+1}d x \right )}{a} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)/(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))/(sin(c + d*x) + 1),x))/a