3.1.30 \(\int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx\) [30]

3.1.30.1 Optimal result

Integrand size = 25, antiderivative size = 279 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=\frac {e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}+\frac {e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}+\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d} \]

output

1/2*e^(3/2)*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a^2/d+1/4*e^(3/2)*arctan( 
1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/a^2/d*2^(1/2)-1/4*e^(3/2)*arctan(1 
+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/a^2/d*2^(1/2)-1/8*e^(3/2)*ln(e^(1/2 
)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/a^2/d*2^(1/2)+1/8*e^(3/ 
2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/a^2/d*2^(1/ 
2)-1/2*e*(e*cot(d*x+c))^(1/2)/d/(a^2+a^2*cot(d*x+c))
 

3.1.30.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.56 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=\frac {5 \left (-2 e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+\left (-e^2\right )^{3/4} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )-\left (-e^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )+2 e \sqrt {e \cot (c+d x)}\right )-\frac {2 (e \cot (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},-\cot (c+d x)\right )}{e}}{10 a^2 d} \]

input

Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x])^2,x]
 

output

(5*(-2*e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]] + (-e^2)^(3/4)*ArcTan[ 
Sqrt[e*Cot[c + d*x]]/(-e^2)^(1/4)] - (-e^2)^(3/4)*ArcTanh[Sqrt[e*Cot[c + d 
*x]]/(-e^2)^(1/4)] + 2*e*Sqrt[e*Cot[c + d*x]]) - (2*(e*Cot[c + d*x])^(5/2) 
*Hypergeometric2F1[2, 5/2, 7/2, -Cot[c + d*x]])/e)/(10*a^2*d)
 

3.1.30.3 Rubi [A] (warning: unable to verify)

Time = 1.22 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.87, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.880, Rules used = {3042, 4050, 27, 3042, 4136, 27, 2030, 3042, 3957, 266, 826, 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.

\(\displaystyle \int \frac {(e \cot (c+d x))^{3/2}}{(a \cot (c+d x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 4050

\(\displaystyle -\frac {\int \frac {-a \cot ^2(c+d x) e^2+a e^2-2 a \cot (c+d x) e^2}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-a \cot ^2(c+d x) e^2+a e^2-2 a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {-a \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^2+a e^2+2 a \tan \left (c+d x+\frac {\pi }{2}\right ) e^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 4136

\(\displaystyle -\frac {\frac {\int -\frac {4 a^2 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}+a e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx-2 e^2 \int \frac {\cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 2030

\(\displaystyle -\frac {a e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx-2 e \int \sqrt {e \cot (c+d x)}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-2 e \int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 e^2 \int \frac {\sqrt {e \cot (c+d x)}}{\cot ^2(c+d x) e^2+e^2}d(e \cot (c+d x))}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \int \frac {e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 826

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \int \frac {e^2 \cot ^2(c+d x)+e}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} \int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}\right )-\frac {1}{2} \int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {\frac {a e^2 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {e^2 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {2 e \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {2 e^{3/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}+\frac {4 e^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )}\)

input

Int[(e*Cot[c + d*x])^(3/2)/(a + a*Cot[c + d*x])^2,x]
 

output

-1/2*(e*Sqrt[e*Cot[c + d*x]])/(d*(a^2 + a^2*Cot[c + d*x])) - ((2*e^(3/2)*A 
rcTan[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 + (Log[e - Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + 
d*x]^2]/(2*Sqrt[2]*Sqrt[e]) - Log[e + Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*C 
ot[c + d*x]^2]/(2*Sqrt[2]*Sqrt[e]))/2))/d)/(4*a^2)
 

3.1.30.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 216

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])
 

rule 217

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])
 

rule 266

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]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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]]))
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 2030

Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m   Int[(b*v) 
^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3957

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]
 

rule 4050

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 
 1)*((c + d*Tan[e + f*x])^(n - 1)/(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 - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 
*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[1, n, 2] && IntegerQ[ 
2*m]
 

rule 4117

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]
 

rule 4136

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]
 

3.1.30.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(-\frac {2 e^{3} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e \left (e^{2}\right )^{\frac {1}{4}}}-\frac {-\frac {\sqrt {e \cot \left (d x +c \right )}}{2 \left (e \cot \left (d x +c \right )+e \right )}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e}\right )}{d \,a^{2}}\)	\(197\)
default	\(-\frac {2 e^{3} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e \left (e^{2}\right )^{\frac {1}{4}}}-\frac {-\frac {\sqrt {e \cot \left (d x +c \right )}}{2 \left (e \cot \left (d x +c \right )+e \right )}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e}\right )}{d \,a^{2}}\)	\(197\)

input

int((e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^2,x,method=_RETURNVERBOSE)
 

output

-2/d/a^2*e^3*(1/16/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))-1/2/e* 
(-1/2*(e*cot(d*x+c))^(1/2)/(e*cot(d*x+c)+e)+1/2/e^(1/2)*arctan((e*cot(d*x+ 
c))^(1/2)/e^(1/2))))
 

3.1.30.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 1172, normalized size of antiderivative = 4.20 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=\text {Too large to display} \]

input

integrate((e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^2,x, algorithm="fricas")
 

output

[1/4*((e*cos(2*d*x + 2*c) + e*sin(2*d*x + 2*c) + e)*sqrt(-e)*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)) - 2*e*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)*(-e^6/( 
a^8*d^4))^(1/4)*log(a^6*d^3*(-e^6/(a^8*d^4))^(3/4) + e^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^6/(a^8*d^4))^(1/4)*log(I*a^6*d^3*(-e^6/(a^8*d 
^4))^(3/4) + e^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^6/(a^8*d^4)) 
^(1/4)*log(-I*a^6*d^3*(-e^6/(a^8*d^4))^(3/4) + e^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^6/(a^8*d^4))^(1/4)*log(-a^6*d^3*(-e^6/(a^8*d^4))^(3/4) + e 
^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), 1/4*(2*(e*cos(2*d*x + 2*c) + e*sin(2 
*d*x + 2*c) + e)*sqrt(e)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 
2*c))/sqrt(e)) - 2*e*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)*(- 
e^6/(a^8*d^4))^(1/4)*log(a^6*d^3*(-e^6/(a^8*d^4))^(3/4) + e^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...
 

3.1.30.6 Sympy [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=\frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\cot ^{2}{\left (c + d x \right )} + 2 \cot {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

input

integrate((e*cot(d*x+c))**(3/2)/(a+a*cot(d*x+c))**2,x)
 

output

Integral((e*cot(c + d*x))**(3/2)/(cot(c + d*x)**2 + 2*cot(c + d*x) + 1), x 
)/a**2
 

3.1.30.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

input

integrate((e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^2,x, algorithm="maxima")
 

output

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
 

3.1.30.8 Giac [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate((e*cot(d*x+c))^(3/2)/(a+a*cot(d*x+c))^2,x, algorithm="giac")
 

output

integrate((e*cot(d*x + c))^(3/2)/(a*cot(d*x + c) + a)^2, x)
 

3.1.30.9 Mupad [B] (verification not implemented)

Time = 13.44 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.35 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx=-\frac {\mathrm {atan}\left (\frac {4\,e^{16}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{a^8\,d^4}\right )}^{1/4}}{\frac {4\,e^{18}}{a^2\,d}+4\,a^2\,d\,e^{15}\,\sqrt {-\frac {e^6}{a^8\,d^4}}}+\frac {4\,e^{13}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{a^8\,d^4}\right )}^{3/4}}{\frac {4\,e^{18}}{a^6\,d^3}+\frac {4\,e^{15}\,\sqrt {-\frac {e^6}{a^8\,d^4}}}{a^2\,d}}\right )\,{\left (-\frac {e^6}{a^8\,d^4}\right )}^{1/4}}{2}-\mathrm {atan}\left (\frac {e^{16}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{256\,a^8\,d^4}\right )}^{1/4}\,16{}\mathrm {i}}{\frac {4\,e^{18}}{a^2\,d}-64\,a^2\,d\,e^{15}\,\sqrt {-\frac {e^6}{256\,a^8\,d^4}}}-\frac {e^{13}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{256\,a^8\,d^4}\right )}^{3/4}\,256{}\mathrm {i}}{\frac {4\,e^{18}}{a^6\,d^3}-\frac {64\,e^{15}\,\sqrt {-\frac {e^6}{256\,a^8\,d^4}}}{a^2\,d}}\right )\,{\left (-\frac {e^6}{256\,a^8\,d^4}\right )}^{1/4}\,2{}\mathrm {i}-\frac {e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\left (a^2\,d\,e+a^2\,d\,e\,\mathrm {cot}\left (c+d\,x\right )\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {-e^3}\,1{}\mathrm {i}}{e^2}\right )\,\sqrt {-e^3}\,1{}\mathrm {i}}{2\,a^2\,d} \]

input

int((e*cot(c + d*x))^(3/2)/(a + a*cot(c + d*x))^2,x)
 

output

(atan(((e*cot(c + d*x))^(1/2)*(-e^3)^(1/2)*1i)/e^2)*(-e^3)^(1/2)*1i)/(2*a^ 
2*d) - atan((e^16*(e*cot(c + d*x))^(1/2)*(-e^6/(256*a^8*d^4))^(1/4)*16i)/( 
(4*e^18)/(a^2*d) - 64*a^2*d*e^15*(-e^6/(256*a^8*d^4))^(1/2)) - (e^13*(e*co 
t(c + d*x))^(1/2)*(-e^6/(256*a^8*d^4))^(3/4)*256i)/((4*e^18)/(a^6*d^3) - ( 
64*e^15*(-e^6/(256*a^8*d^4))^(1/2))/(a^2*d)))*(-e^6/(256*a^8*d^4))^(1/4)*2 
i - (e^2*(e*cot(c + d*x))^(1/2))/(2*(a^2*d*e + a^2*d*e*cot(c + d*x))) - (a 
tan((4*e^16*(e*cot(c + d*x))^(1/2)*(-e^6/(a^8*d^4))^(1/4))/((4*e^18)/(a^2* 
d) + 4*a^2*d*e^15*(-e^6/(a^8*d^4))^(1/2)) + (4*e^13*(e*cot(c + d*x))^(1/2) 
*(-e^6/(a^8*d^4))^(3/4))/((4*e^18)/(a^6*d^3) + (4*e^15*(-e^6/(a^8*d^4))^(1 
/2))/(a^2*d)))*(-e^6/(a^8*d^4))^(1/4))/2