Integrand size = 25, antiderivative size = 278 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}+\frac {\sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {\sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac {\sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}-\frac {\sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d} \]
1/2*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a^2/d+1/4*arctan(1-2^(1/2 )*(e*cot(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a^2/d*2^(1/2)-1/4*arctan(1+2^(1/2) *(e*cot(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a^2/d*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))*e^(1/2)/a^2/d*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))*e^(1/2)/a^2/d*2^(1/ 2)+1/2*(e*cot(d*x+c))^(1/2)/d/(a^2+a^2*cot(d*x+c))
Time = 2.20 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=-\frac {\sqrt {e} \left (-4 \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-2 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+2 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-\frac {4 \sqrt {e \cot (c+d x)}}{\sqrt {e} (1+\cot (c+d x))}-\sqrt {2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+\sqrt {2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )\right )}{8 a^2 d} \]
-1/8*(Sqrt[e]*(-4*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]] - 2*Sqrt[2]*ArcTan[ 1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] + 2*Sqrt[2]*ArcTan[1 + (Sqrt[2 ]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - (4*Sqrt[e*Cot[c + d*x]])/(Sqrt[e]*(1 + Cot[c + d*x])) - Sqrt[2]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt [e*Cot[c + d*x]]] + Sqrt[2]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*S qrt[e*Cot[c + d*x]]]))/(a^2*d)
Time = 1.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.90, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 4051, 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.
| \(\displaystyle \int \frac {\sqrt {e \cot (c+d x)}}{(a \cot (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4051 |
| \(\displaystyle \frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}-\frac {\int -\frac {-a e \cot ^2(c+d x)+2 a e \cot (c+d x)+a e}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-a e \cot ^2(c+d x)+2 a e \cot (c+d x)+a e}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-a e \tan \left (c+d x+\frac {\pi }{2}\right )^2-2 a e \tan \left (c+d x+\frac {\pi }{2}\right )+a e}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\int \frac {4 a^2 e}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}-a e \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \int \frac {1}{\sqrt {e \cot (c+d x)}}dx-a e \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e \int \frac {1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {2 e^2 \int \frac {1}{\sqrt {e \cot (c+d x)} \left (\cot ^2(c+d x) e^2+e^2\right )}d(e \cot (c+d x))}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \int \frac {1}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e^2 \cot ^2(c+d x)+e}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-a e \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 e^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {a e \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 e^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 e^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}-\frac {4 e^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-\frac {4 e^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {2 \sqrt {e} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}}{4 a^2}+\frac {\sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\) |
Sqrt[e*Cot[c + d*x]]/(2*d*(a^2 + a^2*Cot[c + d*x])) + ((-2*Sqrt[e]*ArcTan[ Cot[c + d*x]/Sqrt[e]])/d - (4*e^2*((-(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)/(4*a^2)
3.1.31.3.1 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*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2 )) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c *(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && Int egerQ[2*m]
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[(((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 = 200, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(-\frac {2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e^{3}}-\frac {\frac {\sqrt {e \cot \left (d x +c \right )}}{2 e \left (e \cot \left (d x +c \right )+e \right )}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {3}{2}}}}{2 e}\right )}{d \,a^{2}}\) | \(200\) |
| default | \(-\frac {2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e^{3}}-\frac {\frac {\sqrt {e \cot \left (d x +c \right )}}{2 e \left (e \cot \left (d x +c \right )+e \right )}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {3}{2}}}}{2 e}\right )}{d \,a^{2}}\) | \(200\) |
-2/d/a^2*e^3*(1/16/e^3*(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))-1/2/ e*(1/2*(e*cot(d*x+c))^(1/2)/e/(e*cot(d*x+c)+e)+1/2/e^(3/2)*arctan((e*cot(d *x+c))^(1/2)/e^(1/2))))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 1114, normalized size of antiderivative = 4.01 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=\text {Too large to display} \]
[1/4*(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*cos(2*d*x + 2*c) + a^2*d*sin(2*d*x + 2*c) + a^2*d)*(-e^2/(a ^8*d^4))^(1/4)*log(a^2*d*(-e^2/(a^8*d^4))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) - (I*a^2*d*cos(2*d*x + 2*c) + I*a^2*d*sin(2*d*x + 2*c) + I*a^2*d)*(-e^2/(a^8*d^4))^(1/4)*log(I*a^2*d*(-e^2/(a^8*d^4))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) - (-I*a^2*d*cos(2*d*x + 2*c) - I*a^2*d*sin(2*d*x + 2*c) - I*a^2*d)*(-e^2/(a^8*d^4))^(1/4)*log(-I *a^2*d*(-e^2/(a^8*d^4))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) + (a^2*d*cos(2*d*x + 2*c) + a^2*d*sin(2*d*x + 2*c) + a^2*d)*(-e^2/( a^8*d^4))^(1/4)*log(-a^2*d*(-e^2/(a^8*d^4))^(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))*sin(2*d*x + 2*c))/(a^2*d*cos(2*d*x + 2*c) + a^2*d*sin(2*d*x + 2*c) + a^2*d), 1/4*(2*sqrt(e)*(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)*arctan(sq rt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) - (a^2*d*cos(2*d*x + 2*c) + a^2*d*sin(2*d*x + 2*c) + a^2*d)*(-e^2/(a^8*d^4))^(1/4)*log(a^2*d* (-e^2/(a^8*d^4))^(1/4) + sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) - (I*a^2*d*cos(2*d*x + 2*c) + I*a^2*d*sin(2*d*x + 2*c) + I*a^2*d)*(-e^2/(a ^8*d^4))^(1/4)*log(I*a^2*d*(-e^2/(a^8*d^4))^(1/4) + sqrt((e*cos(2*d*x +...
\[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=\frac {\int \frac {\sqrt {e \cot {\left (c + d x \right )}}}{\cot ^{2}{\left (c + d x \right )} + 2 \cot {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Exception generated. \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=\int { \frac {\sqrt {e \cot \left (d x + c\right )}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Time = 13.13 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^2} \, dx=\frac {\mathrm {atan}\left (\frac {4\,e^{12}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^2}{a^8\,d^4}\right )}^{1/4}}{\frac {4\,e^{13}}{a^2\,d}-4\,a^2\,d\,e^{12}\,\sqrt {-\frac {e^2}{a^8\,d^4}}}+\frac {4\,e^{11}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^2}{a^8\,d^4}\right )}^{3/4}}{\frac {4\,e^{13}}{a^6\,d^3}-\frac {4\,e^{12}\,\sqrt {-\frac {e^2}{a^8\,d^4}}}{a^2\,d}}\right )\,{\left (-\frac {e^2}{a^8\,d^4}\right )}^{1/4}}{2}+\mathrm {atan}\left (\frac {e^{12}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^2}{256\,a^8\,d^4}\right )}^{1/4}\,16{}\mathrm {i}}{\frac {4\,e^{13}}{a^2\,d}+64\,a^2\,d\,e^{12}\,\sqrt {-\frac {e^2}{256\,a^8\,d^4}}}-\frac {e^{11}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^2}{256\,a^8\,d^4}\right )}^{3/4}\,256{}\mathrm {i}}{\frac {4\,e^{13}}{a^6\,d^3}+\frac {64\,e^{12}\,\sqrt {-\frac {e^2}{256\,a^8\,d^4}}}{a^2\,d}}\right )\,{\left (-\frac {e^2}{256\,a^8\,d^4}\right )}^{1/4}\,2{}\mathrm {i}+\frac {e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\left (a^2\,d\,e+a^2\,d\,e\,\mathrm {cot}\left (c+d\,x\right )\right )}-\frac {\sqrt {-e}\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {-e}}\right )\,1{}\mathrm {i}}{2\,a^2\,d} \]
(atan((4*e^12*(e*cot(c + d*x))^(1/2)*(-e^2/(a^8*d^4))^(1/4))/((4*e^13)/(a^ 2*d) - 4*a^2*d*e^12*(-e^2/(a^8*d^4))^(1/2)) + (4*e^11*(e*cot(c + d*x))^(1/ 2)*(-e^2/(a^8*d^4))^(3/4))/((4*e^13)/(a^6*d^3) - (4*e^12*(-e^2/(a^8*d^4))^ (1/2))/(a^2*d)))*(-e^2/(a^8*d^4))^(1/4))/2 + atan((e^12*(e*cot(c + d*x))^( 1/2)*(-e^2/(256*a^8*d^4))^(1/4)*16i)/((4*e^13)/(a^2*d) + 64*a^2*d*e^12*(-e ^2/(256*a^8*d^4))^(1/2)) - (e^11*(e*cot(c + d*x))^(1/2)*(-e^2/(256*a^8*d^4 ))^(3/4)*256i)/((4*e^13)/(a^6*d^3) + (64*e^12*(-e^2/(256*a^8*d^4))^(1/2))/ (a^2*d)))*(-e^2/(256*a^8*d^4))^(1/4)*2i + (e*(e*cot(c + d*x))^(1/2))/(2*(a ^2*d*e + a^2*d*e*cot(c + d*x))) - ((-e)^(1/2)*atan(((e*cot(c + d*x))^(1/2) *1i)/(-e)^(1/2))*1i)/(2*a^2*d)