Integrand size = 31, antiderivative size = 205 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {9 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {a} d}-\frac {9 \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}} \] Output:
-9/128*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(1/2)/d-9/128* cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-3/64*cot(d*x+c)*csc(d*x+c)/d/(a+a*sin( d*x+c))^(1/2)+29/80*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c))^(1/2)+1/40* cot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)-1/5*cot(d*x+c)*csc(d*x+c) ^4/d/(a+a*sin(d*x+c))^(1/2)
Time = 2.75 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.00 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\csc ^{15}\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (820 \cos \left (\frac {1}{2} (c+d x)\right )+1600 \cos \left (\frac {3}{2} (c+d x)\right )+1616 \cos \left (\frac {5}{2} (c+d x)\right )-30 \cos \left (\frac {7}{2} (c+d x)\right )+90 \cos \left (\frac {9}{2} (c+d x)\right )-820 \sin \left (\frac {1}{2} (c+d x)\right )+450 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-450 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+1600 \sin \left (\frac {3}{2} (c+d x)\right )-1616 \sin \left (\frac {5}{2} (c+d x)\right )-225 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+225 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-30 \sin \left (\frac {7}{2} (c+d x)\right )-90 \sin \left (\frac {9}{2} (c+d x)\right )+45 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-45 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{640 d \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^5 \sqrt {a (1+\sin (c+d x))}} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]
Output:
-1/640*(Csc[(c + d*x)/2]^15*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(820*Cos [(c + d*x)/2] + 1600*Cos[(3*(c + d*x))/2] + 1616*Cos[(5*(c + d*x))/2] - 30 *Cos[(7*(c + d*x))/2] + 90*Cos[(9*(c + d*x))/2] - 820*Sin[(c + d*x)/2] + 4 50*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 450*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 1600*Sin[(3*(c + d*x) )/2] - 1616*Sin[(5*(c + d*x))/2] - 225*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 225*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/ 2]]*Sin[3*(c + d*x)] - 30*Sin[(7*(c + d*x))/2] - 90*Sin[(9*(c + d*x))/2] + 45*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 45*Log [1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^5*Sqrt[a*(1 + Sin[c + d*x])])
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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a \sin (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^6 \sqrt {a \sin (c+d x)+a}}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \frac {\csc ^6(c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {\csc ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3258 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\int \frac {\csc (c+d x) (a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {a-a \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-2 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-2 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle -\frac {\frac {4 a \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{2 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^6 \sqrt {\sin (c+d x) a+a}}dx-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int -\frac {\csc ^5(c+d x) (11 \sin (c+d x) a+a)}{2 \sqrt {\sin (c+d x) a+a}}dx}{5 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\csc ^5(c+d x) (11 \sin (c+d x) a+a)}{\sqrt {\sin (c+d x) a+a}}dx}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {11 \sin (c+d x) a+a}{\sin (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^4(c+d x) \left (7 \sin (c+d x) a^2+87 a^2\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^4(c+d x) \left (7 \sin (c+d x) a^2+87 a^2\right )}{\sqrt {\sin (c+d x) a+a}}dx}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {7 \sin (c+d x) a^2+87 a^2}{\sin (c+d x)^4 \sqrt {\sin (c+d x) a+a}}dx}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {\frac {\int -\frac {15 \csc ^3(c+d x) \left (3 a^3-29 a^3 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {5 \int \frac {\csc ^3(c+d x) \left (3 a^3-29 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {5 \int \frac {3 a^3-29 a^3 \sin (c+d x)}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (\frac {\int -\frac {\csc ^2(c+d x) \left (119 a^4-9 a^4 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {\int \frac {\csc ^2(c+d x) \left (119 a^4-9 a^4 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {\int \frac {119 a^4-9 a^4 \sin (c+d x)}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {\frac {\int -\frac {\csc (c+d x) \left (137 a^5-119 a^5 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {-\frac {\int \frac {\csc (c+d x) \left (137 a^5-119 a^5 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {-\frac {\int \frac {137 a^5-119 a^5 \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {-\frac {137 a^4 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-256 a^5 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {-\frac {137 a^4 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-256 a^5 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {-\frac {137 a^4 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {512 a^5 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {-\frac {5 \left (-\frac {-\frac {137 a^4 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {256 \sqrt {2} a^{9/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {119 a^4 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{2 a}-\frac {29 a^2 \cot (c+d x) \csc ^2(c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]
Output:
$Aborted
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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[(-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] - Simp[1/(2* b*(n + 1)*(c^2 - d^2)) Int[(c + d*Sin[e + f*x])^(n + 1)*(Simp[a*d - 2*b*c *(n + 1) + b*d*(2*n + 3)*Sin[e + f*x], x]/Sqrt[a + b*Sin[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] && LtQ[n, -1] && IntegerQ[2*n]
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 = 0.38 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (45 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {5}{2}}-210 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {7}{2}}+45 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{7} \sin \left (d x +c \right )^{5}+128 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {9}{2}}+210 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {11}{2}}-45 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {13}{2}}\right )}{640 a^{\frac {15}{2}} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(180\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBO SE)
Output:
-1/640*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(15/2)*(45*(-a*(sin(d*x+ c)-1))^(9/2)*a^(5/2)-210*(-a*(sin(d*x+c)-1))^(7/2)*a^(7/2)+45*arctanh((-a* (sin(d*x+c)-1))^(1/2)/a^(1/2))*a^7*sin(d*x+c)^5+128*(-a*(sin(d*x+c)-1))^(5 /2)*a^(9/2)+210*(-a*(sin(d*x+c)-1))^(3/2)*a^(11/2)-45*(-a*(sin(d*x+c)-1))^ (1/2)*a^(13/2))/sin(d*x+c)^5/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. 472 vs. \(2 (177) = 354\).
Time = 0.10 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.30 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {45 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - {\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} + \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 (45 \, \cos \left (d x + c\right )^{5} + 15 \, \cos \left (d x + c\right )^{4} + 142 \, \cos \left (d x + c\right )^{3} + 186 \, \cos \left (d x + c\right )^{2} - {\left (45 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{3} + 172 \, \cos \left (d x + c\right )^{2} - 14 \, \cos \left (d x + c\right ) - 73\right )} \sin \left (d x + c\right ) - 59 \, \cos \left (d x + c\right ) - 73\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2560 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d - {\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\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/2),x, algorithm="f ricas")
Output:
1/2560*(45*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (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) + 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*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*(45*cos (d*x + c)^5 + 15*cos(d*x + c)^4 + 142*cos(d*x + c)^3 + 186*cos(d*x + c)^2 - (45*cos(d*x + c)^4 + 30*cos(d*x + c)^3 + 172*cos(d*x + c)^2 - 14*cos(d*x + c) - 73)*sin(d*x + c) - 59*cos(d*x + c) - 73)*sqrt(a*sin(d*x + c) + a)) /(a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d - (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*c os(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c))
\[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**2/(a+a*sin(d*x+c))**(1/2),x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)**2/sqrt(a*(sin(c + d*x) + 1)), x)
\[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/2),x, algorithm="m axima")
Output:
integrate(cot(d*x + c)^4*csc(d*x + c)^2/sqrt(a*sin(d*x + c) + a), x)
Time = 0.19 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\frac {45 \, \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {45 \, \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (720 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1680 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 512 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 420 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{1280 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/2),x, algorithm="g iac")
Output:
1/1280*(45*log(abs(1/2*sqrt(2) + 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))) - 45*log(abs(-1/2*sqrt(2) + 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)*(720*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 1680*sqrt(a)*sin( -1/4*pi + 1/2*d*x + 1/2*c)^7 + 512*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^ 5 + 420*sqrt(a)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 45*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)^5*a*sgn(cos( -1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2}{{\sin \left (c+d\,x\right )}^6\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)^2*(a + a*sin(c + d*x))^(1/2)),x)
Output:
int((sin(c + d*x)^2 - 1)^2/(sin(c + d*x)^6*(a + a*sin(c + d*x))^(1/2)), x)
\[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2}}{\sin \left (d x +c \right )+1}d x \right )}{a} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^2/(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)**2)/(sin (c + d*x) + 1),x))/a