Integrand size = 25, antiderivative size = 306 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\frac {5 \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d e^{3/2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{3/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d e^{3/2}}+\frac {5}{2 a^2 d e \sqrt {e \cot (c+d x)}}-\frac {1}{2 d e \sqrt {e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{3/2}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d e^{3/2}} \]
5/2*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a^2/d/e^(3/2)-1/4*arctan(1-2^(1/2 )*(e*cot(d*x+c))^(1/2)/e^(1/2))/a^2/d/e^(3/2)*2^(1/2)+1/4*arctan(1+2^(1/2) *(e*cot(d*x+c))^(1/2)/e^(1/2))/a^2/d/e^(3/2)*2^(1/2)-1/8*ln(e^(1/2)+cot(d* x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/a^2/d/e^(3/2)*2^(1/2)+1/8*ln(e^ (1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/a^2/d/e^(3/2)*2^(1/ 2)+5/2/a^2/d/e/(e*cot(d*x+c))^(1/2)-1/2/d/e/(a^2+a^2*cot(d*x+c))/(e*cot(d* x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.17 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\frac {8 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\cot (c+d x)\right )+8 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},-\cot (c+d x)\right )+\sqrt {2} \sqrt {e \cot (c+d x)} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )\right )}{8 a^2 d e^{3/2} \sqrt {e \cot (c+d x)}} \]
(8*Sqrt[e]*Hypergeometric2F1[-1/2, 1, 1/2, -Cot[c + d*x]] + 8*Sqrt[e]*Hype rgeometric2F1[-1/2, 2, 1/2, -Cot[c + d*x]] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]* (-2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] + 2*ArcTan[1 + (Sqr t[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]] + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt [2]*Sqrt[e*Cot[c + d*x]]]))/(8*a^2*d*e^(3/2)*Sqrt[e*Cot[c + d*x]])
Time = 1.57 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.94, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 27, 73, 216}
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.
\(\displaystyle \int \frac {1}{(a \cot (c+d x)+a)^2 (e \cot (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4052 |
\(\displaystyle -\frac {\int -\frac {3 e \cot ^2(c+d x) a^2+5 e a^2-2 e \cot (c+d x) a^2}{2 (e \cot (c+d x))^{3/2} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {3 e \cot ^2(c+d x) a^2+5 e a^2-2 e \cot (c+d x) a^2}{(e \cot (c+d x))^{3/2} (\cot (c+d x) a+a)}dx}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {3 e \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^2+5 e a^2+2 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {\frac {2 \int -\frac {7 a^3 e^3+5 a^3 \cot ^2(c+d x) e^3+2 a^3 \cot (c+d x) e^3}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}+\frac {10 a}{d \sqrt {e \cot (c+d x)}}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {7 a^3 e^3+5 a^3 \cot ^2(c+d x) e^3+2 a^3 \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {7 a^3 e^3+5 a^3 \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^3-2 a^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {\int \frac {4 a^4 e^3}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+2 a^2 e^3 \int \frac {1}{\sqrt {e \cot (c+d x)}}dx}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+2 a^2 e^3 \int \frac {1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {2 a^2 e^4 \int \frac {1}{\sqrt {e \cot (c+d x)} \left (\cot ^2(c+d x) e^2+e^2\right )}d(e \cot (c+d x))}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \int \frac {1}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e^2 \cot ^2(c+d x)+e}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 a^2 e^4 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {\frac {5 a^3 e^3 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 a^2 e^4 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {\frac {5 a^2 e^3 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 a^2 e^4 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {-\frac {10 a^2 e^2 \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}-\frac {4 a^2 e^4 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {10 a}{d \sqrt {e \cot (c+d x)}}-\frac {\frac {10 a^2 e^{5/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}-\frac {4 a^2 e^4 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{a e^3}}{4 a^3 e}-\frac {1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt {e \cot (c+d x)}}\) |
-1/2*1/(d*e*Sqrt[e*Cot[c + d*x]]*(a^2 + a^2*Cot[c + d*x])) + ((10*a)/(d*Sq rt[e*Cot[c + d*x]]) - ((10*a^2*e^(5/2)*ArcTan[Cot[c + d*x]/Sqrt[e]])/d - ( 4*a^2*e^4*((-(ArcTan[1 - Sqrt[2]*Sqrt[e]*Cot[c + d*x]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + Sqrt[2]*Sqrt[e]*Cot[c + d*x]]/(Sqrt[2]*Sqrt[e]))/(2*e) + (-1/ 2*Log[e - Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(Sqrt[2]*Sqrt [e]) + Log[e + Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(2*Sqrt[ 2]*Sqrt[e]))/(2*e)))/d)/(a*e^3))/(4*a^3*e)
3.1.33.3.1 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, 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[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
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_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + 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, B, 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 = 212, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(-\frac {2 e^{3} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e^{5}}-\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}-\frac {\frac {\sqrt {e \cot \left (d x +c \right )}}{2 e \cot \left (d x +c \right )+2 e}+\frac {5 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e^{4}}\right )}{d \,a^{2}}\) | \(212\) |
default | \(-\frac {2 e^{3} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e^{5}}-\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}-\frac {\frac {\sqrt {e \cot \left (d x +c \right )}}{2 e \cot \left (d x +c \right )+2 e}+\frac {5 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e^{4}}\right )}{d \,a^{2}}\) | \(212\) |
-2/d/a^2*e^3*(-1/16/e^5*(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/e ^4/(e*cot(d*x+c))^(1/2)-1/2/e^4*(1/2*(e*cot(d*x+c))^(1/2)/(e*cot(d*x+c)+e) +5/2/e^(1/2)*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))))
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 1257, normalized size of antiderivative = 4.11 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\text {Too large to display} \]
[-1/4*(5*sqrt(-e)*(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)*log((e*cos(2*d *x + 2*c) - e*sin(2*d*x + 2*c) - 2*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/ sin(2*d*x + 2*c))*sin(2*d*x + 2*c) + e)/(cos(2*d*x + 2*c) + sin(2*d*x + 2* c) + 1)) - (a^2*d*e^2*cos(2*d*x + 2*c) + a^2*d*e^2*sin(2*d*x + 2*c) + a^2* d*e^2)*(-1/(a^8*d^4*e^6))^(1/4)*log(a^2*d*e^2*(-1/(a^8*d^4*e^6))^(1/4) + s qrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) + (-I*a^2*d*e^2*cos(2*d*x + 2*c) - I*a^2*d*e^2*sin(2*d*x + 2*c) - I*a^2*d*e^2)*(-1/(a^8*d^4*e^6))^(1 /4)*log(I*a^2*d*e^2*(-1/(a^8*d^4*e^6))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) + (I*a^2*d*e^2*cos(2*d*x + 2*c) + I*a^2*d*e^2*sin(2* d*x + 2*c) + I*a^2*d*e^2)*(-1/(a^8*d^4*e^6))^(1/4)*log(-I*a^2*d*e^2*(-1/(a ^8*d^4*e^6))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) + (a ^2*d*e^2*cos(2*d*x + 2*c) + a^2*d*e^2*sin(2*d*x + 2*c) + a^2*d*e^2)*(-1/(a ^8*d^4*e^6))^(1/4)*log(-a^2*d*e^2*(-1/(a^8*d^4*e^6))^(1/4) + sqrt((e*cos(2 *d*x + 2*c) + e)/sin(2*d*x + 2*c))) + 2*sqrt((e*cos(2*d*x + 2*c) + e)/sin( 2*d*x + 2*c))*(4*cos(2*d*x + 2*c) - 5*sin(2*d*x + 2*c) - 4))/(a^2*d*e^2*co s(2*d*x + 2*c) + a^2*d*e^2*sin(2*d*x + 2*c) + a^2*d*e^2), 1/4*(10*sqrt(e)* (cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) + (a^2*d*e^2*cos(2*d*x + 2*c) + a^2*d*e^2* sin(2*d*x + 2*c) + a^2*d*e^2)*(-1/(a^8*d^4*e^6))^(1/4)*log(a^2*d*e^2*(-1/( a^8*d^4*e^6))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) ...
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )} + 2 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{2}} \]
Integral(1/((e*cot(c + d*x))**(3/2)*cot(c + d*x)**2 + 2*(e*cot(c + d*x))** (3/2)*cot(c + d*x) + (e*cot(c + d*x))**(3/2)), x)/a**2
Exception generated. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\int { \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{2} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 13.24 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx=\frac {\frac {5\,\mathrm {cot}\left (c+d\,x\right )}{2}+2}{a^2\,d\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}+a^2\,d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}-\frac {\mathrm {atan}\left (\frac {2048\,a^{10}\,d^5\,e^{13}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{a^8\,d^4\,e^6}\right )}^{1/4}}{51200\,a^8\,d^4\,e^{12}-2048\,a^{12}\,d^6\,e^{15}\,\sqrt {-\frac {1}{a^8\,d^4\,e^6}}}+\frac {51200\,a^{14}\,d^7\,e^{16}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{a^8\,d^4\,e^6}\right )}^{3/4}}{51200\,a^8\,d^4\,e^{12}-2048\,a^{12}\,d^6\,e^{15}\,\sqrt {-\frac {1}{a^8\,d^4\,e^6}}}\right )\,{\left (-\frac {1}{a^8\,d^4\,e^6}\right )}^{1/4}}{2}-\mathrm {atan}\left (\frac {a^{10}\,d^5\,e^{13}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^4\,e^6}\right )}^{1/4}\,8192{}\mathrm {i}}{51200\,a^8\,d^4\,e^{12}+32768\,a^{12}\,d^6\,e^{15}\,\sqrt {-\frac {1}{256\,a^8\,d^4\,e^6}}}-\frac {a^{14}\,d^7\,e^{16}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^4\,e^6}\right )}^{3/4}\,3276800{}\mathrm {i}}{51200\,a^8\,d^4\,e^{12}+32768\,a^{12}\,d^6\,e^{15}\,\sqrt {-\frac {1}{256\,a^8\,d^4\,e^6}}}\right )\,{\left (-\frac {1}{256\,a^8\,d^4\,e^6}\right )}^{1/4}\,2{}\mathrm {i}+\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {-e^3}\,1{}\mathrm {i}}{e^2}\right )\,\sqrt {-e^3}\,5{}\mathrm {i}}{2\,a^2\,d\,e^3} \]
((5*cot(c + d*x))/2 + 2)/(a^2*d*(e*cot(c + d*x))^(3/2) + a^2*d*e*(e*cot(c + d*x))^(1/2)) - (atan((2048*a^10*d^5*e^13*(e*cot(c + d*x))^(1/2)*(-1/(a^8 *d^4*e^6))^(1/4))/(51200*a^8*d^4*e^12 - 2048*a^12*d^6*e^15*(-1/(a^8*d^4*e^ 6))^(1/2)) + (51200*a^14*d^7*e^16*(e*cot(c + d*x))^(1/2)*(-1/(a^8*d^4*e^6) )^(3/4))/(51200*a^8*d^4*e^12 - 2048*a^12*d^6*e^15*(-1/(a^8*d^4*e^6))^(1/2) ))*(-1/(a^8*d^4*e^6))^(1/4))/2 - atan((a^10*d^5*e^13*(e*cot(c + d*x))^(1/2 )*(-1/(256*a^8*d^4*e^6))^(1/4)*8192i)/(51200*a^8*d^4*e^12 + 32768*a^12*d^6 *e^15*(-1/(256*a^8*d^4*e^6))^(1/2)) - (a^14*d^7*e^16*(e*cot(c + d*x))^(1/2 )*(-1/(256*a^8*d^4*e^6))^(3/4)*3276800i)/(51200*a^8*d^4*e^12 + 32768*a^12* d^6*e^15*(-1/(256*a^8*d^4*e^6))^(1/2)))*(-1/(256*a^8*d^4*e^6))^(1/4)*2i + (atan(((e*cot(c + d*x))^(1/2)*(-e^3)^(1/2)*1i)/e^2)*(-e^3)^(1/2)*5i)/(2*a^ 2*d*e^3)