Integrand size = 25, antiderivative size = 325 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \left (a^2+b^2\right ) d e^{3/2}}-\frac {(a+b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{3/2}}+\frac {(a+b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{3/2}}+\frac {2}{a d e \sqrt {e \cot (c+d x)}}+\frac {(a-b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d e^{3/2}}-\frac {(a-b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d e^{3/2}} \]
2*b^(5/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/a^(3/2)/(a^ 2+b^2)/d/e^(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^(3/2)*2^(1/2)+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^(3/2)*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^(3/2)*2^(1/2)-1/4*(a-b)*l n(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^( 3/2)*2^(1/2)+2/a/d/e/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.46 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\frac {8 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {b \cot (c+d x)}{a}\right )+a \left (8 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\sqrt {2} b \sqrt {\cot (c+d x)} \left (-2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{4 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}} \]
(8*b^2*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] + a*(8*a*Hyp ergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*b*Sqrt[Cot[c + d*x ]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[ Cot[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + Log[ 1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(4*a*(a^2 + b^2)*d*e*Sqr t[e*Cot[c + d*x]])
Time = 1.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.91, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4052, 27, 3042, 4136, 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.
| \(\displaystyle \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {2 \int -\frac {b \cot ^2(c+d x) e^2+b e^2+a \cot (c+d x) e^2}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e^3}+\frac {2}{a d e \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\int \frac {b \cot ^2(c+d x) e^2+b e^2+a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\int \frac {b \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^2+b e^2-a \tan \left (c+d x+\frac {\pi }{2}\right ) e^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {\int \frac {a b e^2+a^2 \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {b^3 e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {\int \frac {a b e^2-a^2 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {2 \int -\frac {a e^2 (b e+a \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 \int \frac {a e^2 (b e+a \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \int \frac {b e+a \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^2 \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {-\frac {2 b^3 e \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {2 b^{5/2} e^{3/2} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {2 a e^2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}\) |
2/(a*d*e*Sqrt[e*Cot[c + d*x]]) - ((2*b^(5/2)*e^(3/2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*(a^2 + b^2)*d) - (2*a*e^2*(((a + b)* (-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])))/2 - ((a - b)*(-1/2*Log[e + e*Cot[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]*Sqrt[e])))/2))/((a^2 + b^2)*d))/(a*e^3)
3.1.73.3.1 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[(((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 = 354, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(-\frac {2 e^{2} \left (-\frac {b^{3} \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{a \,e^{3} \left (a^{2}+b^{2}\right ) \sqrt {a e b}}+\frac {-\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right ) e^{3}}-\frac {1}{a \,e^{3} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(354\) |
| default | \(-\frac {2 e^{2} \left (-\frac {b^{3} \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{a \,e^{3} \left (a^{2}+b^{2}\right ) \sqrt {a e b}}+\frac {-\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right ) e^{3}}-\frac {1}{a \,e^{3} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(354\) |
-2/d*e^2*(-1/a/e^3*b^3/(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^3*(-1/8*b/e*(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))-1/8*a/(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/a /e^3/(e*cot(d*x+c))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1798 vs. \(2 (262) = 524\).
Time = 0.41 (sec) , antiderivative size = 3637, normalized size of antiderivative = 11.19 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\text {Too large to display} \]
[1/2*(((a^3 + a*b^2)*d*e^2*cos(2*d*x + 2*c) + (a^3 + a*b^2)*d*e^2)*sqrt(-( (a^4 + 2*a^2*b^2 + b^4)*d^2*e^3*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^6)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e^3))*log(-(a^2 - b^2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*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^6)) - (a^2*b - b^3 )*d*e^2)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*e^3*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^6)) + 2*a*b)/(( a^4 + 2*a^2*b^2 + b^4)*d^2*e^3))) - ((a^3 + a*b^2)*d*e^2*cos(2*d*x + 2*c) + (a^3 + a*b^2)*d*e^2)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2*e^3*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^6 )) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e^3))*log(-(a^2 - b^2)*sqrt((e*co s(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*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^6)) - (a^2*b - b^3)*d*e^2)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*d^2* e^3*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^6)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e^3))) - ((a^3 + a*b^2)*d*e^2*cos(2*d*x + 2*c) + (a^3 + a*b^2)*d*e^2)*sqrt(((a^4 + 2*a^2*b^ 2 + b^4)*d^2*e^3*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^6)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2*e...
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cot {\left (c + d x \right )}\right )}\, dx \]
Exception generated. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\int { \frac {1}{{\left (b \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 13.93 (sec) , antiderivative size = 4899, normalized size of antiderivative = 15.07 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx=\text {Too large to display} \]
(log((e*cot(c + d*x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) - ((-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2)*(((((-1/(b^2 *d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2)*(((e*cot(c + d*x))^(1 /2)*(-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2)*(512*a^9* b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^ 3*d^9*e^19))/2 - 512*a^8*b^9*d^8*e^18 - 640*a^10*b^7*d^8*e^18 + 256*a^12*b ^5*d^8*e^18 + 384*a^14*b^3*d^8*e^18))/2 - (e*cot(c + d*x))^(1/2)*(512*a^8* b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 + 128*a^12*b^4*d^7*e^16 + 64*a^14*b^2 *d^7*e^16))*(-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2))/ 2 - 128*a^7*b^8*d^6*e^15 + 32*a^11*b^4*d^6*e^15 + 32*a^13*b^2*d^6*e^15))/2 )*(-1/(b^2*d^2*e^3*1i - a^2*d^2*e^3*1i + 2*a*b*d^2*e^3))^(1/2))/2 - atan(( (-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((e*cot(c + d *x))^(1/2)*(64*a^7*b^7*d^5*e^13 - 32*a^9*b^5*d^5*e^13) - (-1i/(4*(b^2*d^2* e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^ 2*e^3 + a*b*d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b* d^2*e^3*2i)))^(1/2)*((-1i/(4*(b^2*d^2*e^3 - a^2*d^2*e^3 + a*b*d^2*e^3*2i)) )^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^9*b^9*d^9*e^19 + 512*a^11*b^7*d^9*e^ 19 - 512*a^13*b^5*d^9*e^19 - 512*a^15*b^3*d^9*e^19) - 512*a^8*b^9*d^8*e^18 - 640*a^10*b^7*d^8*e^18 + 256*a^12*b^5*d^8*e^18 + 384*a^14*b^3*d^8*e^18) - (e*cot(c + d*x))^(1/2)*(512*a^8*b^8*d^7*e^16 - 448*a^10*b^6*d^7*e^16 ...