3.1.62 \(\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx\) [62]

3.1.62.1 Optimal result

Integrand size = 25, antiderivative size = 372 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx=-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {2 a \left (a^2-3 b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {2 b \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d} \]

output

-2/3*b*(3*a^2-b^2)*(e*cot(d*x+c))^(3/2)/d-32/35*a*b^2*(e*cot(d*x+c))^(5/2) 
/d/e-2/7*b^2*(e*cot(d*x+c))^(5/2)*(a+b*cot(d*x+c))/d/e-1/2*(a-b)*(a^2+4*a* 
b+b^2)*e^(3/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d*2^(1/2)+1/ 
2*(a-b)*(a^2+4*a*b+b^2)*e^(3/2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1 
/2))/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*e^(3/2)*ln(e^(1/2)+cot(d*x+c)*e^( 
1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/d*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*e^( 
3/2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d*2^(1/2) 
-2*a*(a^2-3*b^2)*e*(e*cot(d*x+c))^(1/2)/d
 

3.1.62.2 Mathematica [C] (verified)

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

Time = 2.98 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.67 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx=-\frac {(e \cot (c+d x))^{3/2} \left (\frac {6}{5} a b^2 \cot ^{\frac {5}{2}}(c+d x)+\frac {2}{7} b^3 \cot ^{\frac {7}{2}}(c+d x)+\frac {2}{3} b \left (-3 a^2+b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )+\frac {1}{4} a \left (a^2-3 b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{d \cot ^{\frac {3}{2}}(c+d x)} \]

input

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

output

-(((e*Cot[c + d*x])^(3/2)*((6*a*b^2*Cot[c + d*x]^(5/2))/5 + (2*b^3*Cot[c + 
 d*x]^(7/2))/7 + (2*b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*(-1 + Hypergeometr 
ic2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcT 
an[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot 
[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + 
d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c 
 + d*x]]))/4))/(d*Cot[c + d*x]^(3/2)))
 

3.1.62.3 Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.94, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4049, 27, 3042, 4113, 3042, 4011, 3042, 4011, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4049

\(\displaystyle -\frac {2 \int -\frac {1}{2} (e \cot (c+d x))^{3/2} \left (16 a b^2 e \cot ^2(c+d x)+7 b \left (3 a^2-b^2\right ) e \cot (c+d x)+a \left (7 a^2-5 b^2\right ) e\right )dx}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e \cot (c+d x))^{3/2} \left (16 a b^2 e \cot ^2(c+d x)+7 b \left (3 a^2-b^2\right ) e \cot (c+d x)+a \left (7 a^2-5 b^2\right ) e\right )dx}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (16 a b^2 e \tan \left (c+d x+\frac {\pi }{2}\right )^2-7 b \left (3 a^2-b^2\right ) e \tan \left (c+d x+\frac {\pi }{2}\right )+a \left (7 a^2-5 b^2\right ) e\right )dx}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {\int (e \cot (c+d x))^{3/2} \left (7 a \left (a^2-3 b^2\right ) e+7 b \left (3 a^2-b^2\right ) \cot (c+d x) e\right )dx-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (7 a \left (a^2-3 b^2\right ) e-7 b \left (3 a^2-b^2\right ) e \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {\int \sqrt {e \cot (c+d x)} \left (7 a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)-7 b \left (3 a^2-b^2\right ) e^2\right )dx-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (-7 b \left (3 a^2-b^2\right ) e^2-7 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^2\right )dx-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {\int \frac {-7 a \left (a^2-3 b^2\right ) e^3-7 b \left (3 a^2-b^2\right ) \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)}}dx-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {7 b \left (3 a^2-b^2\right ) e^3 \tan \left (c+d x+\frac {\pi }{2}\right )-7 a \left (a^2-3 b^2\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {\frac {2 \int \frac {7 e^3 \left (a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {14 e^3 \int \frac {a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {14 e^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {14 a e^2 \left (a^2-3 b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {14 b e \left (3 a^2-b^2\right ) (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a b^2 (e \cot (c+d x))^{5/2}}{5 d}}{7 e}-\frac {2 b^2 (e \cot (c+d x))^{5/2} (a+b \cot (c+d x))}{7 d e}\)

input

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

output

(-2*b^2*(e*Cot[c + d*x])^(5/2)*(a + b*Cot[c + d*x]))/(7*d*e) + ((-14*a*(a^ 
2 - 3*b^2)*e^2*Sqrt[e*Cot[c + d*x]])/d - (14*b*(3*a^2 - b^2)*e*(e*Cot[c + 
d*x])^(3/2))/(3*d) - (32*a*b^2*(e*Cot[c + d*x])^(5/2))/(5*d) + (14*e^3*((( 
a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sq 
rt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt 
[e]]/(Sqrt[2]*Sqrt[e])))/2 + ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[e + e* 
Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e]) + L 
og[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2]*S 
qrt[e])))/2))/d)/(7*e)
 

3.1.62.3.1 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 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 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 1482

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]
 

rule 3042

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

rule 4011

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int 
[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 
 0] && GtQ[m, 0]
 

rule 4017

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]
 

rule 4049

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 - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

rule 4113

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + 
 b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si 
mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && 
NeQ[A*b^2 - a*b*B + a^2*C, 0] &&  !LeQ[m, -1]
 

3.1.62.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {3 a e \,b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+a^{2} e^{2} b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}-\frac {b^{3} e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{3} e^{3} \sqrt {e \cot \left (d x +c \right )}-3 \sqrt {e \cot \left (d x +c \right )}\, a \,b^{2} e^{3}-e^{4} \left (\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(410\)
default	\(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {3 a e \,b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+a^{2} e^{2} b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}-\frac {b^{3} e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{3} e^{3} \sqrt {e \cot \left (d x +c \right )}-3 \sqrt {e \cot \left (d x +c \right )}\, a \,b^{2} e^{3}-e^{4} \left (\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(410\)
parts	\(-\frac {2 a^{3} e \left (\sqrt {e \cot \left (d x +c \right )}-\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 )}{8}\right )}{d}-\frac {2 b^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{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 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d \,e^{2}}-\frac {6 a \,b^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\sqrt {e \cot \left (d x +c \right )}\, e^{2}+\frac {e^{2} \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}\right )}{d e}+\frac {3 a^{2} b \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2} \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 )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\)	\(656\)

input

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

output

-2/d/e^2*(1/7*b^3*(e*cot(d*x+c))^(7/2)+3/5*a*e*b^2*(e*cot(d*x+c))^(5/2)+a^ 
2*e^2*b*(e*cot(d*x+c))^(3/2)-1/3*b^3*e^2*(e*cot(d*x+c))^(3/2)+a^3*e^3*(e*c 
ot(d*x+c))^(1/2)-3*(e*cot(d*x+c))^(1/2)*a*b^2*e^3-e^4*(1/8*(a^3*e-3*a*b^2* 
e)*(e^2)^(1/4)/e^2*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*(3*a^2*b-b^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))))
 

3.1.62.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1795 vs. \(2 (311) = 622\).

Time = 0.34 (sec) , antiderivative size = 1795, normalized size of antiderivative = 4.83 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx=\text {Too large to display} \]

input

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

output

-1/210*(105*(d*cos(2*d*x + 2*c) - d)*sqrt(-(2*(3*a^5*b - 10*a^3*b^3 + 3*a* 
b^5)*e^3 + sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4 
*b^8 - 30*a^2*b^10 + b^12)*e^6/d^4)*d^2)/d^2)*log(-(a^12 - 12*a^10*b^2 - 2 
7*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*e^4*sqrt((e*cos(2*d*x + 2*c) 
+ e)/sin(2*d*x + 2*c)) + ((a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3* 
a*b^8)*d*e^3 + (3*a^2*b - b^3)*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 4 
52*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^6/d^4)*d^3)*sqrt(-(2*(3*a 
^5*b - 10*a^3*b^3 + 3*a*b^5)*e^3 + sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 
 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^6/d^4)*d^2)/d^2))*sin 
(2*d*x + 2*c) - 105*(d*cos(2*d*x + 2*c) - d)*sqrt(-(2*(3*a^5*b - 10*a^3*b^ 
3 + 3*a*b^5)*e^3 + sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 
 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^6/d^4)*d^2)/d^2)*log(-(a^12 - 12*a^10 
*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*e^4*sqrt((e*cos(2*d*x 
 + 2*c) + e)/sin(2*d*x + 2*c)) - ((a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3* 
b^6 + 3*a*b^8)*d*e^3 + (3*a^2*b - b^3)*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8 
*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^6/d^4)*d^3)*sqrt( 
-(2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*e^3 + sqrt(-(a^12 - 30*a^10*b^2 + 255 
*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^6/d^4)*d^2)/d 
^2))*sin(2*d*x + 2*c) + 105*(d*cos(2*d*x + 2*c) - d)*sqrt(-(2*(3*a^5*b - 1 
0*a^3*b^3 + 3*a*b^5)*e^3 - sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 45...
 

3.1.62.6 Sympy [F]

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

input

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

output

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

3.1.62.7 Maxima [F(-2)]

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

input

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

3.1.62.8 Giac [F]

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

input

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

output

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

3.1.62.9 Mupad [B] (verification not implemented)

Time = 18.06 (sec) , antiderivative size = 2317, normalized size of antiderivative = 6.23 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3 \, dx=\text {Too large to display} \]

input

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

output

(e*cot(c + d*x))^(3/2)*((2*b^3)/(3*d) - (2*a^2*b)/d) - (e*cot(c + d*x))^(1 
/2)*((2*a^3*e)/d - (6*a*b^2*e)/d) - atan((((16*(e*cot(c + d*x))^(1/2)*(a^6 
*e^6 - b^6*e^6 + 15*a^2*b^4*e^6 - 15*a^4*b^2*e^6))/d^2 - (8*(4*a^3*d^2*e^5 
 - 12*a*b^2*d^2*e^5)*(-(b^6*e^3*1i - a^6*e^3*1i + 6*a*b^5*e^3 + 6*a^5*b*e^ 
3 - a^2*b^4*e^3*15i - 20*a^3*b^3*e^3 + a^4*b^2*e^3*15i)/(4*d^2))^(1/2))/d^ 
3)*(-(b^6*e^3*1i - a^6*e^3*1i + 6*a*b^5*e^3 + 6*a^5*b*e^3 - a^2*b^4*e^3*15 
i - 20*a^3*b^3*e^3 + a^4*b^2*e^3*15i)/(4*d^2))^(1/2)*1i + ((16*(e*cot(c + 
d*x))^(1/2)*(a^6*e^6 - b^6*e^6 + 15*a^2*b^4*e^6 - 15*a^4*b^2*e^6))/d^2 + ( 
8*(4*a^3*d^2*e^5 - 12*a*b^2*d^2*e^5)*(-(b^6*e^3*1i - a^6*e^3*1i + 6*a*b^5* 
e^3 + 6*a^5*b*e^3 - a^2*b^4*e^3*15i - 20*a^3*b^3*e^3 + a^4*b^2*e^3*15i)/(4 
*d^2))^(1/2))/d^3)*(-(b^6*e^3*1i - a^6*e^3*1i + 6*a*b^5*e^3 + 6*a^5*b*e^3 
- a^2*b^4*e^3*15i - 20*a^3*b^3*e^3 + a^4*b^2*e^3*15i)/(4*d^2))^(1/2)*1i)/( 
((16*(e*cot(c + d*x))^(1/2)*(a^6*e^6 - b^6*e^6 + 15*a^2*b^4*e^6 - 15*a^4*b 
^2*e^6))/d^2 + (8*(4*a^3*d^2*e^5 - 12*a*b^2*d^2*e^5)*(-(b^6*e^3*1i - a^6*e 
^3*1i + 6*a*b^5*e^3 + 6*a^5*b*e^3 - a^2*b^4*e^3*15i - 20*a^3*b^3*e^3 + a^4 
*b^2*e^3*15i)/(4*d^2))^(1/2))/d^3)*(-(b^6*e^3*1i - a^6*e^3*1i + 6*a*b^5*e^ 
3 + 6*a^5*b*e^3 - a^2*b^4*e^3*15i - 20*a^3*b^3*e^3 + a^4*b^2*e^3*15i)/(4*d 
^2))^(1/2) - ((16*(e*cot(c + d*x))^(1/2)*(a^6*e^6 - b^6*e^6 + 15*a^2*b^4*e 
^6 - 15*a^4*b^2*e^6))/d^2 - (8*(4*a^3*d^2*e^5 - 12*a*b^2*d^2*e^5)*(-(b^6*e 
^3*1i - a^6*e^3*1i + 6*a*b^5*e^3 + 6*a^5*b*e^3 - a^2*b^4*e^3*15i - 20*a...