Integrand size = 29, antiderivative size = 153 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {23 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 a^{5/2} d}+\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}+\frac {9 \cot (c+d x)}{4 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d \sqrt {a+a \sin (c+d x)}} \]
-23/4*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d+4*arcta nh(1/2*cos(d*x+c)*a^(1/2)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2 )+9/4*cot(d*x+c)/a^2/d/(a+a*sin(d*x+c))^(1/2)-1/2*cot(d*x+c)*csc(d*x+c)/a^ 2/d/(a+a*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 2.82 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.02 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (-40-(256+256 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right )+20 \cot \left (\frac {1}{4} (c+d x)\right )-\csc ^2\left (\frac {1}{4} (c+d x)\right )-92 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+92 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{4} (c+d x)\right )+\frac {2}{\left (\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}-\frac {40 \sin \left (\frac {1}{4} (c+d x)\right )}{\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )}-\frac {2}{\left (\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}+\frac {40 \sin \left (\frac {1}{4} (c+d x)\right )}{\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )}+20 \tan \left (\frac {1}{4} (c+d x)\right )\right )}{32 d (a (1+\sin (c+d x)))^{5/2}} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(-40 - (256 + 256*I)*(-1)^(3/4)*A rcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 20*Cot[(c + d*x)/ 4] - Csc[(c + d*x)/4]^2 - 92*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 92*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2 /(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (40*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (40*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 20*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x]))^(5/2))
Time = 2.03 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.88, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {3042, 3359, 3042, 3258, 3042, 3464, 3042, 3128, 219, 3252, 219, 3523, 27, 3042, 3463, 27, 3042, 3464, 3042, 3128, 219, 3252, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a \sin (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^3 (a \sin (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3359 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (\sin ^2(c+d x)+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \int \frac {\csc ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \int \frac {1}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3258 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \frac {\csc (c+d x) (a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \frac {a-a \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-2 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-2 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\frac {4 a \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^2+1}{\sin (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^2(c+d x) (a-7 a \sin (c+d x))}{2 \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^2(c+d x) (a-7 a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {a-7 a \sin (c+d x)}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {\csc (c+d x) \left (15 a^2-a^2 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\csc (c+d x) \left (15 a^2-a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {15 a^2-a^2 \sin (c+d x)}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {-\frac {-\frac {15 a \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-16 a^2 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {15 a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-16 a^2 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {32 a^2 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+15 a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {15 a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {16 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {16 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {30 a^2 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {16 \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {30 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {\cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (-\frac {\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
(-2*(-1/2*((-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d *x]]])/d + (2*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt [a + a*Sin[c + d*x]])])/d)/a - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]]))) /a^2 + (-1/2*(Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (- 1/2*((-30*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]] )/d + (16*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/d)/a - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/ (4*a))/a^2
3.5.87.3.1 Defintions of rubi rules used
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[-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[((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 = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (23 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{3} \left (\sin ^{2}\left (d x +c \right )\right )-16 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (\sin ^{2}\left (d x +c \right )\right )+9 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-7 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {5}{2}}\right )}{4 a^{\frac {11}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(164\) |
-1/4/a^(11/2)*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(23*arctanh((-a*(si n(d*x+c)-1))^(1/2)/a^(1/2))*a^3*sin(d*x+c)^2-16*2^(1/2)*arctanh(1/2*(-a*(s in(d*x+c)-1))^(1/2)*2^(1/2)/a^(1/2))*a^3*sin(d*x+c)^2+9*(-a*(sin(d*x+c)-1) )^(3/2)*a^(3/2)-7*(-a*(sin(d*x+c)-1))^(1/2)*a^(5/2))/sin(d*x+c)^2/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. 508 vs. \(2 (128) = 256\).
Time = 0.31 (sec) , antiderivative size = 508, normalized size of antiderivative = 3.32 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {23 \, {\left (\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 )} \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 ) + \frac {32 \, \sqrt {2} {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - 4 \, {\left (9 \, \cos \left (d x + c\right )^{2} + {\left (9 \, \cos \left (d x + c\right ) + 11\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 11\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
1/16*(23*(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)*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)) + 32*sqrt(2)*(a *cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) + (a*cos(d*x + c)^2 - a)*sin(d*x + c) - a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c ) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/s qrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - 4*(9*cos(d*x + c)^2 + (9*cos(d*x + c) + 11)*sin(d*x + c) - 2*cos(d*x + c) - 11)*sqrt(a*sin(d*x + c) + a))/(a^3*d *cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 - a^3*d*cos(d*x + c) - a^3*d + (a^3 *d*cos(d*x + c)^2 - a^3*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 0.43 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.46 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {23 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {32 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {32 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4 \, {\left (18 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{16 \, d} \]
-1/16*sqrt(2)*sqrt(a)*(23*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^3*sgn (cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 32*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 32*log(-sin(-1/4*pi + 1/2 *d*x + 1/2*c) + 1)/(a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 4*(18*sin(- 1/4*pi + 1/2*d*x + 1/2*c)^3 - 7*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)^2*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) )/d
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]