Integrand size = 25, antiderivative size = 351 \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right ) d e^{5/2}}-\frac {(a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {(a-b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d e^{5/2}} \]
-2*b^(7/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/a^(5/2)/(a ^2+b^2)/d/e^(5/2)+2/3/a/d/e/(e*cot(d*x+c))^(3/2)-1/2*(a-b)*arctan(1-2^(1/2 )*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)/d/e^(5/2)*2^(1/2)+1/2*(a-b)*arct an(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)/d/e^(5/2)*2^(1/2)-1/4 *(a+b)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^ 2)/d/e^(5/2)*2^(1/2)+1/4*(a+b)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*co t(d*x+c))^(1/2))/(a^2+b^2)/d/e^(5/2)*2^(1/2)-2*b/a^2/d/e^2/(e*cot(d*x+c))^ (1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.31 \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=\frac {2 \left (b^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {b \cot (c+d x)}{a}\right )+a \left (a \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )-3 b \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )\right )\right )}{3 a \left (a^2+b^2\right ) d e (e \cot (c+d x))^{3/2}} \]
(2*(b^2*Hypergeometric2F1[-3/2, 1, -1/2, -((b*Cot[c + d*x])/a)] + a*(a*Hyp ergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] - 3*b*Cot[c + d*x]*Hypergeom etric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])))/(3*a*(a^2 + b^2)*d*e*(e*Cot[c + d*x])^(3/2))
Time = 1.69 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.94, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.920, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4137, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
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.
| \(\displaystyle \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {2 \int -\frac {3 \left (b \cot ^2(c+d x) e^2+b e^2+a \cot (c+d x) e^2\right )}{2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}dx}{3 a e^3}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\int \frac {b \cot ^2(c+d x) e^2+b e^2+a \cot (c+d x) e^2}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}dx}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\int \frac {b \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^2+b e^2-a \tan \left (c+d x+\frac {\pi }{2}\right ) e^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {2 \int \frac {\left (a^2-b^2\right ) e^4-b^2 e^4 \cot ^2(c+d x)}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {\int \frac {\left (a^2-b^2\right ) e^4-b^2 e^4 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {\int \frac {\left (a^2-b^2\right ) e^4-b^2 e^4 \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 4137 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {a^3 e^4-a^2 b e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}-\frac {b^4 e^4 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {a^3 e^4+a^2 b \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {\frac {2 \int -\frac {a^2 e^4 (a e-b e \cot (c+d x))}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 \int \frac {a^2 e^4 (a e-b e \cot (c+d x))}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \int \frac {a e-b e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {b^4 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a^2 e^4 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {b^4 e^4 \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 a^2 e^4 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {\frac {2 b^4 e^3 \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {2 a^2 e^4 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {\frac {-\frac {2 a^2 e^4 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^{7/2} e^{7/2} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{a e^3}+\frac {2 b e}{a d \sqrt {e \cot (c+d x)}}}{a e^3}\) |
2/(3*a*d*e*(e*Cot[c + d*x])^(3/2)) - ((2*b*e)/(a*d*Sqrt[e*Cot[c + d*x]]) + ((-2*b^(7/2)*e^(7/2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(S qrt[a]*(a^2 + b^2)*d) - (2*a^2*e^4*(((a - b)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e *Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*C ot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])))/2 + ((a + b)*(-1/2*Log[e + e*Co t[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e]) + Log [e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqr t[e])))/2))/((a^2 + b^2)*d))/(a*e^3))/(a*e^3)
3.1.74.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Sim p[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*C)*T an[e + f*x], x], x], x] + Simp[(A*b^2 + a^2*C)/(a^2 + b^2) Int[(c + d*Tan [e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{ a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.04 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(-\frac {2 e^{2} \left (\frac {b^{4} \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{a^{2} e^{4} \left (a^{2}+b^{2}\right ) \sqrt {a e b}}+\frac {-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right ) e^{4}}-\frac {1}{3 a \,e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b}{a^{2} e^{4} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(371\) |
| default | \(-\frac {2 e^{2} \left (\frac {b^{4} \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{a^{2} e^{4} \left (a^{2}+b^{2}\right ) \sqrt {a e b}}+\frac {-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right ) e^{4}}-\frac {1}{3 a \,e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b}{a^{2} e^{4} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(371\) |
-2/d*e^2*(1/a^2/e^4*b^4/(a^2+b^2)/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2 )*b/(a*e*b)^(1/2))+1/(a^2+b^2)/e^4*(-1/8*a/e*(e^2)^(1/4)*2^(1/2)*(ln((e*co t(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+ c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2) /(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot( d*x+c))^(1/2)+1))+1/8*b/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)* (e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot (d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d *x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/ 3/a/e^3/(e*cot(d*x+c))^(3/2)+1/a^2/e^4*b/(e*cot(d*x+c))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1849 vs. \(2 (284) = 568\).
Time = 0.47 (sec) , antiderivative size = 3742, normalized size of antiderivative = 10.66 \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=\text {Too large to display} \]
[-1/6*(3*((a^4 + a^2*b^2)*d*e^3*cos(2*d*x + 2*c) + (a^4 + a^2*b^2)*d*e^3)* sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) + 2*a*b)/((a^4 + 2*a ^2*b^2 + b^4)*d^2*e^5))*log(-(a^2 - b^2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin (2*d*x + 2*c)) + ((a^4*b + 2*a^2*b^3 + b^5)*d^3*e^8*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) + (a^3 - a*b^2)*d*e^3)*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*e^5*sqrt(-(a^4 - 2*a^2* b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) + 2 *a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e^5))) - 3*((a^4 + a^2*b^2)*d*e^3*cos(2 *d*x + 2*c) + (a^4 + a^2*b^2)*d*e^3)*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*e^5 *sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e^5))*log(-(a^2 - b ^2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - ((a^4*b + 2*a^2*b^3 + b^5)*d^3*e^8*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) + (a^3 - a*b^2)*d*e^3)*sqrt(((a^4 + 2*a^2*b ^2 + b^4)*d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4* b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e^ 5))) - 3*((a^4 + a^2*b^2)*d*e^3*cos(2*d*x + 2*c) + (a^4 + a^2*b^2)*d*e^3)* sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*e^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4*e^10)) - 2*a*b)/((a^4 +...
\[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \left (a + b \cot {\left (c + d x \right )}\right )}\, dx \]
Exception generated. \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=\int { \frac {1}{{\left (b \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Time = 14.96 (sec) , antiderivative size = 6042, normalized size of antiderivative = 17.21 \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx=\text {Too large to display} \]
(2/(3*a*e) - (2*b*cot(c + d*x))/(a^2*e))/(d*(e*cot(c + d*x))^(3/2)) - atan (((((e*cot(c + d*x))^(1/2)*(64*a^14*b^9*d^5*e^18 + 32*a^18*b^5*d^5*e^18))/ 2 + ((1/(b^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2*a*b*d^2*e^5))^(1/2)*(((((1/(b ^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2*a*b*d^2*e^5))^(1/2)*(((e*cot(c + d*x))^ (1/2)*(1/(b^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2*a*b*d^2*e^5))^(1/2)*(512*a^1 8*b^9*d^9*e^28 + 512*a^20*b^7*d^9*e^28 - 512*a^22*b^5*d^9*e^28 - 512*a^24* b^3*d^9*e^28))/4 - 256*a^16*b^10*d^8*e^26 - 256*a^18*b^8*d^8*e^26 + 192*a^ 20*b^6*d^8*e^26 + 128*a^22*b^4*d^8*e^26 - 64*a^24*b^2*d^8*e^26))/2 - ((e*c ot(c + d*x))^(1/2)*(512*a^15*b^10*d^7*e^23 + 448*a^19*b^6*d^7*e^23 - 128*a ^21*b^4*d^7*e^23 - 64*a^23*b^2*d^7*e^23))/2)*(1/(b^2*d^2*e^5*1i - a^2*d^2* e^5*1i + 2*a*b*d^2*e^5))^(1/2))/2 + 192*a^15*b^9*d^6*e^21 - 16*a^19*b^5*d^ 6*e^21 - 16*a^21*b^3*d^6*e^21))/2)*(1/(b^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2 *a*b*d^2*e^5))^(1/2)*1i + (((e*cot(c + d*x))^(1/2)*(64*a^14*b^9*d^5*e^18 + 32*a^18*b^5*d^5*e^18))/2 + ((1/(b^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2*a*b*d ^2*e^5))^(1/2)*(((((1/(b^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2*a*b*d^2*e^5))^( 1/2)*(((e*cot(c + d*x))^(1/2)*(1/(b^2*d^2*e^5*1i - a^2*d^2*e^5*1i + 2*a*b* d^2*e^5))^(1/2)*(512*a^18*b^9*d^9*e^28 + 512*a^20*b^7*d^9*e^28 - 512*a^22* b^5*d^9*e^28 - 512*a^24*b^3*d^9*e^28))/4 + 256*a^16*b^10*d^8*e^26 + 256*a^ 18*b^8*d^8*e^26 - 192*a^20*b^6*d^8*e^26 - 128*a^22*b^4*d^8*e^26 + 64*a^24* b^2*d^8*e^26))/2 - ((e*cot(c + d*x))^(1/2)*(512*a^15*b^10*d^7*e^23 + 44...