Integrand size = 31, antiderivative size = 329 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {1587 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{16384 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d} \] Output:
-1587/16384*a^(3/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d-1 587/16384*a^2*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-529/8192*a^2*cot(d*x+c)* csc(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-529/10240*a^2*cot(d*x+c)*csc(d*x+c)^2/ d/(a+a*sin(d*x+c))^(1/2)+8653/35840*a^2*cot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin (d*x+c))^(1/2)+1957/4480*a^2*cot(d*x+c)*csc(d*x+c)^4/d/(a+a*sin(d*x+c))^(1 /2)+83/448*a^2*cot(d*x+c)*csc(d*x+c)^5/d/(a+a*sin(d*x+c))^(1/2)-3/112*a*co t(d*x+c)*csc(d*x+c)^6*(a+a*sin(d*x+c))^(1/2)/d-1/8*cot(d*x+c)*csc(d*x+c)^7 *(a+a*sin(d*x+c))^(3/2)/d
Time = 7.67 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.84 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]
Output:
-1/573440*(a*Csc[(c + d*x)/2]^25*Sqrt[a*(1 + Sin[c + d*x])]*(3037258*Cos[( c + d*x)/2] + 10394286*Cos[(3*(c + d*x))/2] + 3369650*Cos[(5*(c + d*x))/2] + 3171574*Cos[(7*(c + d*x))/2] - 2341070*Cos[(9*(c + d*x))/2] + 866502*Co s[(11*(c + d*x))/2] - 37030*Cos[(13*(c + d*x))/2] - 111090*Cos[(15*(c + d* x))/2] + 1944075*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3110520*Co s[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1555260*Cos[ 4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 444360*Cos[6*( c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 55545*Cos[8*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1944075*Log[1 - Cos[( c + d*x)/2] + Sin[(c + d*x)/2]] + 3110520*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 1555260*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 444360*Cos[6*(c + d*x)]*Log[1 - Cos[(c + d*x )/2] + Sin[(c + d*x)/2]] - 55545*Cos[8*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3037258*Sin[(c + d*x)/2] + 10394286*Sin[(3*(c + d*x ))/2] - 3369650*Sin[(5*(c + d*x))/2] + 3171574*Sin[(7*(c + d*x))/2] + 2341 070*Sin[(9*(c + d*x))/2] + 866502*Sin[(11*(c + d*x))/2] + 37030*Sin[(13*(c + d*x))/2] - 111090*Sin[(15*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Cs c[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^8)
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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^{3/2}}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \csc ^9(c+d x) (\sin (c+d x) a+a)^{3/2} \left (1-2 \sin ^2(c+d x)\right )dx+\int \csc ^5(c+d x) (\sin (c+d x) a+a)^{3/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)^5}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3241 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx-\frac {1}{4} a \int -\frac {15}{2} \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {15}{8} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \left (\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \left (\frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^9}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {3}{2} \csc ^8(c+d x) (a-7 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{8 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \csc ^8(c+d x) (a-7 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {(a-7 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)^8}dx}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {3 \left (\frac {1}{7} \int -\frac {1}{2} \csc ^7(c+d x) \sqrt {\sin (c+d x) a+a} \left (87 \sin (c+d x) a^2+83 a^2\right )dx-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {1}{14} \int \csc ^7(c+d x) \sqrt {\sin (c+d x) a+a} \left (87 \sin (c+d x) a^2+83 a^2\right )dx-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (-\frac {1}{14} \int \frac {\sqrt {\sin (c+d x) a+a} \left (87 \sin (c+d x) a^2+83 a^2\right )}{\sin (c+d x)^7}dx-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \int \csc ^6(c+d x) \sqrt {\sin (c+d x) a+a}dx\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^6}dx\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \int \csc ^5(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^5}dx-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \left (\frac {7}{8} \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \left (\frac {7}{8} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {1}{14} \left (\frac {83 a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {1957}{12} a^2 \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}\right )\right )-\frac {a^2 \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}\right )}{16 a}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {15}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b *Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], 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] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 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[((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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], 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] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 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 = 0.49 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (55545 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {15}{2}} a^{\frac {7}{2}}-425845 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {9}{2}}+1418249 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {11}{2}}-55545 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{11} \sin \left (d x +c \right )^{8}-2509197 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {13}{2}}+2176627 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {15}{2}}-416759 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {17}{2}}-425845 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {19}{2}}+55545 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {21}{2}}\right )}{573440 a^{\frac {19}{2}} \sin \left (d x +c \right )^{8} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(234\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBO SE)
Output:
1/573440*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(19/2)*(55545*(-a*(sin (d*x+c)-1))^(15/2)*a^(7/2)-425845*(-a*(sin(d*x+c)-1))^(13/2)*a^(9/2)+14182 49*(-a*(sin(d*x+c)-1))^(11/2)*a^(11/2)-55545*arctanh((-a*(sin(d*x+c)-1))^( 1/2)/a^(1/2))*a^11*sin(d*x+c)^8-2509197*(-a*(sin(d*x+c)-1))^(9/2)*a^(13/2) +2176627*(-a*(sin(d*x+c)-1))^(7/2)*a^(15/2)-416759*(-a*(sin(d*x+c)-1))^(5/ 2)*a^(17/2)-425845*(-a*(sin(d*x+c)-1))^(3/2)*a^(19/2)+55545*(-a*(sin(d*x+c )-1))^(1/2)*a^(21/2))/sin(d*x+c)^8/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. 657 vs. \(2 (289) = 578\).
Time = 0.12 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.00 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x, algorithm="f ricas")
Output:
1/2293760*(55545*(a*cos(d*x + c)^9 + a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^7 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^5 + 6*a*cos(d*x + c)^4 - 4*a*cos( d*x + c)^3 - 4*a*cos(d*x + c)^2 + a*cos(d*x + c) + (a*cos(d*x + c)^8 - 4*a *cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 + a)*sin(d*x + c ) + a)*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*(55545*a*cos(d*x + c)^8 + 37 030*a*cos(d*x + c)^7 - 214774*a*cos(d*x + c)^6 + 27358*a*cos(d*x + c)^5 + 199004*a*cos(d*x + c)^4 - 185006*a*cos(d*x + c)^3 - 153786*a*cos(d*x + c)^ 2 + 48938*a*cos(d*x + c) + (55545*a*cos(d*x + c)^7 + 18515*a*cos(d*x + c)^ 6 - 196259*a*cos(d*x + c)^5 - 223617*a*cos(d*x + c)^4 - 24613*a*cos(d*x + c)^3 + 160393*a*cos(d*x + c)^2 + 6607*a*cos(d*x + c) - 42331*a)*sin(d*x + c) + 42331*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^9 + d*cos(d*x + c) ^8 - 4*d*cos(d*x + c)^7 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^5 + 6*d*co s(d*x + c)^4 - 4*d*cos(d*x + c)^3 - 4*d*cos(d*x + c)^2 + d*cos(d*x + c) + (d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c) + d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**5*(a+a*sin(d*x+c))**(3/2),x)
Output:
Timed out
\[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{5} \,d x } \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x, algorithm="m axima")
Output:
integrate((a*sin(d*x + c) + a)^(3/2)*cot(d*x + c)^4*csc(d*x + c)^5, x)
Time = 0.19 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.03 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x, algorithm="g iac")
Output:
-1/2293760*sqrt(2)*(55545*sqrt(2)*a*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(co s(-1/4*pi + 1/2*d*x + 1/2*c)) + 4*(7109760*a*sgn(cos(-1/4*pi + 1/2*d*x + 1 /2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^15 - 27254080*a*sgn(cos(-1/4*pi + 1/ 2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^13 + 45383968*a*sgn(cos(-1/ 4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^11 - 40147152*a*sg n(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 + 17413 016*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 1667036*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1 /2*c)^5 - 851690*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d *x + 1/2*c)^3 + 55545*a*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)^8)*sqrt(a)/d
Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^9} \,d x \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^5,x)
Output:
int(((sin(c + d*x)^2 - 1)^2*(a + a*sin(c + d*x))^(3/2))/sin(c + d*x)^9, x)
\[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{5} \sin \left (d x +c \right )d x +\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{5}d x \right ) \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4*csc(c + d*x)**5*sin( c + d*x),x) + int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4*csc(c + d*x)**5,x ))