Integrand size = 29, antiderivative size = 182 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 a^{3/2} d}-\frac {3 \cot (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{32 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 a^2 d} \] Output:
-3/64*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(3/2)/d-3/64*co t(d*x+c)/a/d/(a+a*sin(d*x+c))^(1/2)-1/32*cot(d*x+c)*csc(d*x+c)/a/d/(a+a*si n(d*x+c))^(1/2)+5/8*cot(d*x+c)*csc(d*x+c)^2/a/d/(a+a*sin(d*x+c))^(1/2)-1/4 *cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2)/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(376\) vs. \(2(182)=364\).
Time = 2.55 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.07 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\csc ^{12}\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (446 \cos \left (\frac {1}{2} (c+d x)\right )-182 \cos \left (\frac {3}{2} (c+d x)\right )-2 \cos \left (\frac {5}{2} (c+d x)\right )-6 \cos \left (\frac {7}{2} (c+d x)\right )+9 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-12 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-446 \sin \left (\frac {1}{2} (c+d x)\right )-182 \sin \left (\frac {3}{2} (c+d x)\right )+2 \sin \left (\frac {5}{2} (c+d x)\right )-6 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{64 d \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4 (a (1+\sin (c+d x)))^{3/2}} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]
Output:
-1/64*(Csc[(c + d*x)/2]^12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(446*Co s[(c + d*x)/2] - 182*Cos[(3*(c + d*x))/2] - 2*Cos[(5*(c + d*x))/2] - 6*Cos [(7*(c + d*x))/2] + 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*Co s[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 9*Log[1 - Cos[(c + d *x)/2] + Sin[(c + d*x)/2]] + 12*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 446*Sin[(c + d*x)/2] - 182*Sin[(3*(c + d*x))/2] + 2*Sin[(5*( c + d*x))/2] - 6*Sin[(7*(c + d*x))/2]))/(d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4*(a*(1 + Sin[c + d*x]))^(3/2))
Time = 2.43 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.89, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {3042, 3359, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3252, 219, 3523, 27, 3042, 3459, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a \sin (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^5 (a \sin (c+d x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3359 |
| \(\displaystyle \frac {\int \csc ^5(c+d x) \sqrt {\sin (c+d x) a+a} \left (\sin ^2(c+d x)+1\right )dx}{a^2}-\frac {2 \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^5}dx}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\frac {\int \frac {1}{2} \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a} (13 \sin (c+d x) a+a)dx}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a} (13 \sin (c+d x) a+a)dx}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin (c+d x) a+a} (13 \sin (c+d x) a+a)}{\sin (c+d x)^4}dx}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {83}{6} a \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}}{8 a}-\frac {\cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}}{a^2}-\frac {2 \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]
Output:
(-2*(-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))/a^2 + (-1/4*(Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/d + (-1/3*(a^2*Cot[c + d*x ]*Csc[c + d*x]^2)/(d*Sqrt[a + a*Sin[c + d*x]]) + (83*a*(-1/2*(a*Cot[c + d* x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (3*(-((Sqrt[a]*ArcTanh[(Sq rt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sq rt[a + a*Sin[c + d*x]])))/4))/6)/(8*a))/a^2
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[-2/(a*b*d) Int[(d* Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 2), x], x] + Simp[1/a^2 I nt[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2), x] , x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]
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 = 0.40 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (3 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {5}{2}}-11 a^{\frac {7}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{6} \sin \left (d x +c \right )^{4}-11 a^{\frac {9}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}}+3 a^{\frac {11}{2}} \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\right )}{64 a^{\frac {15}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(162\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE )
Output:
1/64/a^(15/2)*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(3*(-a*(sin(d*x+c)- 1))^(7/2)*a^(5/2)-11*a^(7/2)*(-a*(sin(d*x+c)-1))^(5/2)-3*arctanh((-a*(sin( d*x+c)-1))^(1/2)/a^(1/2))*a^6*sin(d*x+c)^4-11*a^(9/2)*(-a*(sin(d*x+c)-1))^ (3/2)+3*a^(11/2)*(-a*(sin(d*x+c)-1))^(1/2))/sin(d*x+c)^4/cos(d*x+c)/(a+a*s in(d*x+c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (158) = 316\).
Time = 0.10 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.43 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {3 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} + 20 \, \cos \left (d x + c\right )^{2} + {\left (3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 21 \, \cos \left (d x + c\right ) + 39\right )} \sin \left (d x + c\right ) - 18 \, \cos \left (d x + c\right ) - 39\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{256 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right ) + a^{2} d + {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(3/2),x, algorithm="fri cas")
Output:
1/256*(3*(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*(3*cos(d*x + c)^4 + 2*cos(d* x + c)^3 + 20*cos(d*x + c)^2 + (3*cos(d*x + c)^3 + cos(d*x + c)^2 + 21*cos (d*x + c) + 39)*sin(d*x + c) - 18*cos(d*x + c) - 39)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^5 + a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c)^2 + a^2*d*cos(d*x + c) + a^2*d + (a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sin(d*x + c))
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)/(a+a*sin(d*x+c))**(3/2),x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)/(a*(sin(c + d*x) + 1))**(3/2), x)
Timed out. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(3/2),x, algorithm="max ima")
Output:
Timed out
Time = 0.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {4 \, {\left (24 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 44 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 22 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{256 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(3/2),x, algorithm="gia c")
Output:
-1/256*sqrt(2)*sqrt(a)*(3*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)))/(a^2*sgn (cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 4*(24*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 44*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 - 22*sin(-1/4*pi + 1/2*d*x + 1/2*c) ^3 + 3*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^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2}{{\sin \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)*(a + a*sin(c + d*x))^(3/2)),x)
Output:
int((sin(c + d*x)^2 - 1)^2/(sin(c + d*x)^5*(a + a*sin(c + d*x))^(3/2)), x)
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c))^(3/2),x)
Output:
(sqrt(a)*int((sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4*csc(c + d*x))/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1),x))/a**2