3.1.82 \(\int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx\) [82]

3.1.82.1 Optimal result

Integrand size = 25, antiderivative size = 476 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=-\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]

output

-1/4*a^(3/2)*(3*a^4+6*a^2*b^2+35*b^4)*e^(7/2)*arctan(b^(1/2)*(e*cot(d*x+c) 
)^(1/2)/a^(1/2)/e^(1/2))/b^(5/2)/(a^2+b^2)^3/d+1/2*a^2*e^2*(e*cot(d*x+c))^ 
(3/2)/b/(a^2+b^2)/d/(a+b*cot(d*x+c))^2+1/2*(a+b)*(a^2-4*a*b+b^2)*e^(7/2)*a 
rctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(a 
+b)*(a^2-4*a*b+b^2)*e^(7/2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2)) 
/(a^2+b^2)^3/d*2^(1/2)+1/4*(a-b)*(a^2+4*a*b+b^2)*e^(7/2)*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)^3/d*2^(1/2)-1/4*(a-b) 
*(a^2+4*a*b+b^2)*e^(7/2)*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)^3/d*2^(1/2)+1/4*a^2*(3*a^2+11*b^2)*e^3*(e*cot(d*x+c)) 
^(1/2)/b^2/(a^2+b^2)^2/d/(a+b*cot(d*x+c))
 

3.1.82.2 Mathematica [C] (verified)

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

Time = 6.25 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.21 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=-\frac {(e \cot (c+d x))^{7/2} \left (\frac {2 a^{7/2} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right )^3}-\frac {2 a^3 \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{b^2 \left (a^2+b^2\right )^3}+\frac {2 a^2 \left (3 a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right )^3}-\frac {2 a \left (3 a^2-b^2\right ) \cot ^{\frac {5}{2}}(c+d x)}{5 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {7}{2}}(c+d x)}{7 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \left (7 \cot ^{\frac {3}{2}}(c+d x)-3 \cot ^{\frac {7}{2}}(c+d x)-7 \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{21 \left (a^2+b^2\right )^3}+\frac {4 b^2 \cot ^{\frac {9}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {9}{2},\frac {11}{2},-\frac {b \cot (c+d x)}{a}\right )}{9 a \left (a^2+b^2\right )^2}+\frac {2 b^2 \cot ^{\frac {9}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (3,\frac {9}{2},\frac {11}{2},-\frac {b \cot (c+d x)}{a}\right )}{9 a^3 \left (a^2+b^2\right )}-\frac {a \left (a^2-3 b^2\right ) \left (10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+40 \sqrt {\cot (c+d x)}-8 \cot ^{\frac {5}{2}}(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{20 \left (a^2+b^2\right )^3}\right )}{d \cot ^{\frac {7}{2}}(c+d x)} \]

input

Integrate[(e*Cot[c + d*x])^(7/2)/(a + b*Cot[c + d*x])^3,x]
 

output

-(((e*Cot[c + d*x])^(7/2)*((2*a^(7/2)*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[C 
ot[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^3) - (2*a^3*(3*a^2 - b^2)*Sqr 
t[Cot[c + d*x]])/(b^2*(a^2 + b^2)^3) + (2*a^2*(3*a^2 - b^2)*Cot[c + d*x]^( 
3/2))/(3*b*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 
 + b^2)^3) + (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^3) + (2 
*b*(3*a^2 - b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + 
d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^ 
2)^3) + (4*b^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((b*Cot 
[c + d*x])/a)])/(9*a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(9/2)*Hypergeome 
tric2F1[3, 9/2, 11/2, -((b*Cot[c + d*x])/a)])/(9*a^3*(a^2 + b^2)) - (a*(a^ 
2 - 3*b^2)*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 10*Sqrt[2] 
*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 40*Sqrt[Cot[c + d*x]] - 8*Cot[c 
+ d*x]^(5/2) + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x] 
] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(20*(a^ 
2 + b^2)^3)))/(d*Cot[c + d*x]^(7/2)))
 

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

Time = 2.13 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.94, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 4048, 27, 3042, 4128, 27, 3042, 4136, 27, 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.

\(\displaystyle \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4048

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4128

\(\displaystyle \frac {\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac {\int -\frac {\left (3 a^4+3 b^2 a^2+8 b^4\right ) \cot ^2(c+d x) e^4+a^2 \left (3 a^2+11 b^2\right ) e^4-16 a b^3 \cot (c+d x) e^4}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (3 a^4+3 b^2 a^2+8 b^4\right ) \cot ^2(c+d x) e^4+a^2 \left (3 a^2+11 b^2\right ) e^4-16 a b^3 \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (3 a^4+3 b^2 a^2+8 b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^4+a^2 \left (3 a^2+11 b^2\right ) e^4+16 a b^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {\frac {\int \frac {8 \left (a b^2 \left (a^2-3 b^2\right ) e^4-b^3 \left (3 a^2-b^2\right ) e^4 \cot (c+d x)\right )}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {8 \int \frac {a b^2 \left (a^2-3 b^2\right ) e^4-b^3 \left (3 a^2-b^2\right ) e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {8 \int \frac {a b^2 \left (a^2-3 b^2\right ) e^4+b^3 \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {\frac {\frac {16 \int -\frac {b^2 e^4 \left (a \left (a^2-3 b^2\right ) e-b \left (3 a^2-b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int \frac {b^2 e^4 \left (a \left (a^2-3 b^2\right ) e-b \left (3 a^2-b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \int \frac {a \left (a^2-3 b^2\right ) e-b \left (3 a^2-b^2\right ) e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {-\frac {2 a^2 e^3 \left (3 a^4+6 a^2 b^2+35 b^4\right ) \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}+\frac {\frac {2 a^{3/2} e^{7/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}-\frac {16 b^2 e^4 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

input

Int[(e*Cot[c + d*x])^(7/2)/(a + b*Cot[c + d*x])^3,x]
 

output

(a^2*e^2*(e*Cot[c + d*x])^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2 
) + ((a^2*(3*a^2 + 11*b^2)*e^3*Sqrt[e*Cot[c + d*x]])/(b*(a^2 + b^2)*d*(a + 
 b*Cot[c + d*x])) + ((2*a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*e^(7/2)*ArcTa 
n[(Sqrt[b]*Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)*d) - (16 
*b^2*e^4*(((a + b)*(a^2 - 4*a*b + b^2)*(-(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)*(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]) + 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))/(2*b*(a^2 + b^2)))/(4*b*(a^2 + 
b^2))
 

3.1.82.3.1 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 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 218

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]
 

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 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 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
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] && GtQ[m, 2] && LtQ 
[n, -1] && 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 4128

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

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]
 

3.1.82.4 Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(-\frac {2 e^{4} \left (-\frac {a^{2} \left (\frac {\frac {\left (5 a^{4}+18 a^{2} b^{2}+13 b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 b}+\frac {a e \left (3 a^{4}+14 a^{2} b^{2}+11 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 b^{2}}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}-\frac {\left (3 a^{4}+6 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{8 b^{2} \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3}}\right )}{d}\)	\(460\)
default	\(-\frac {2 e^{4} \left (-\frac {a^{2} \left (\frac {\frac {\left (5 a^{4}+18 a^{2} b^{2}+13 b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 b}+\frac {a e \left (3 a^{4}+14 a^{2} b^{2}+11 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 b^{2}}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}-\frac {\left (3 a^{4}+6 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{8 b^{2} \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3}}\right )}{d}\)	\(460\)

input

int((e*cot(d*x+c))^(7/2)/(a+b*cot(d*x+c))^3,x,method=_RETURNVERBOSE)
 

output

-2/d*e^4*(-a^2/(a^2+b^2)^3*((1/8*(5*a^4+18*a^2*b^2+13*b^4)/b*(e*cot(d*x+c) 
)^(3/2)+1/8*a*e*(3*a^4+14*a^2*b^2+11*b^4)/b^2*(e*cot(d*x+c))^(1/2))/(e*cot 
(d*x+c)*b+a*e)^2-1/8*(3*a^4+6*a^2*b^2+35*b^4)/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)^3*(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*arc 
tan(-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))))
 

3.1.82.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4483 vs. \(2 (405) = 810\).

Time = 63.17 (sec) , antiderivative size = 9029, normalized size of antiderivative = 18.97 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \]

input

integrate((e*cot(d*x+c))^(7/2)/(a+b*cot(d*x+c))^3,x, algorithm="fricas")
 

output

Too large to include
 

3.1.82.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Timed out} \]

input

integrate((e*cot(d*x+c))**(7/2)/(a+b*cot(d*x+c))**3,x)
 

output

Timed out
 

3.1.82.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

input

integrate((e*cot(d*x+c))^(7/2)/(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
 

3.1.82.8 Giac [F]

\[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

input

integrate((e*cot(d*x+c))^(7/2)/(a+b*cot(d*x+c))^3,x, algorithm="giac")
 

output

integrate((e*cot(d*x + c))^(7/2)/(b*cot(d*x + c) + a)^3, x)
 

3.1.82.9 Mupad [B] (verification not implemented)

Time = 20.49 (sec) , antiderivative size = 20089, normalized size of antiderivative = 42.20 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \]

input

int((e*cot(c + d*x))^(7/2)/(a + b*cot(c + d*x))^3,x)
 

output

(((e*cot(c + d*x))^(1/2)*(3*a^5*e^5 + 11*a^3*b^2*e^5))/(4*b^2*(a^4 + b^4 + 
 2*a^2*b^2)) + (e^4*(e*cot(c + d*x))^(3/2)*(5*a^4 + 13*a^2*b^2))/(4*b*(a^4 
 + b^4 + 2*a^2*b^2)))/(a^2*d*e^2 + b^2*d*e^2*cot(c + d*x)^2 + 2*a*b*d*e^2* 
cot(c + d*x)) - atan(((((32*a*b^18*d^2*e^21 - 18*a^19*d^2*e^21 - 6528*a^3* 
b^16*d^2*e^21 + 2758*a^5*b^14*d^2*e^21 + 26482*a^7*b^12*d^2*e^21 + 21582*a 
^9*b^10*d^2*e^21 + 7594*a^11*b^8*d^2*e^21 + 3314*a^13*b^6*d^2*e^21 + 246*a 
^15*b^4*d^2*e^21 + 90*a^17*b^2*d^2*e^21)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a 
^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^1 
2*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (((1600*a^2*b^23*d^4*e^14 + 1 
2864*a^4*b^21*d^4*e^14 + 45312*a^6*b^19*d^4*e^14 + 91392*a^8*b^17*d^4*e^14 
 + 115584*a^10*b^15*d^4*e^14 + 94080*a^12*b^13*d^4*e^14 + 48384*a^14*b^11* 
d^4*e^14 + 14592*a^16*b^9*d^4*e^14 + 2112*a^18*b^7*d^4*e^14 + 64*a^20*b^5* 
d^4*e^14)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 
 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^ 
16*b^3*d^5) + ((e*cot(c + d*x))^(1/2)*((e^7*1i)/(4*(b^6*d^2 - a^6*d^2 + a* 
b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2* 
d^2)))^(1/2)*(512*b^28*d^4*e^10 + 4608*a^2*b^26*d^4*e^10 + 17920*a^4*b^24* 
d^4*e^10 + 38400*a^6*b^22*d^4*e^10 + 46080*a^8*b^20*d^4*e^10 + 21504*a^10* 
b^18*d^4*e^10 - 21504*a^12*b^16*d^4*e^10 - 46080*a^14*b^14*d^4*e^10 - 3840 
0*a^16*b^12*d^4*e^10 - 17920*a^18*b^10*d^4*e^10 - 4608*a^20*b^8*d^4*e^1...