\(\int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\) [832]

Optimal result

Integrand size = 23, antiderivative size = 296 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \] Output:

-1/2*(a^2-2*a*b-b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b^2) 
^2/d-1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b 
^2)^2/d+a^(5/2)*(3*a^2+7*b^2)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/b^( 
5/2)/(a^2+b^2)^2/d+1/2*(a^2+2*a*b-b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1 
+cot(d*x+c)))*2^(1/2)/(a^2+b^2)^2/d+(3*a^2+2*b^2)/b^2/(a^2+b^2)/d/cot(d*x+ 
c)^(1/2)-a^2/b/(a^2+b^2)/d/cot(d*x+c)^(1/2)/(b+a*cot(d*x+c))
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.39 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {8 a^2 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {a \cot (c+d x)}{b}\right )+4 a^2 \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},-\frac {a \cot (c+d x)}{b}\right )+b^2 \left (-4 \left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\sqrt {2} a b \sqrt {\cot (c+d x)} \left (-2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{2 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}} \] Input:

Integrate[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2),x]
 

Output:

(8*a^2*b^2*Hypergeometric2F1[-1/2, 1, 1/2, -((a*Cot[c + d*x])/b)] + 4*a^2* 
(a^2 + b^2)*Hypergeometric2F1[-1/2, 2, 1/2, -((a*Cot[c + d*x])/b)] + b^2*( 
-4*(a^2 - b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]* 
a*b*Sqrt[Cot[c + d*x]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcT 
an[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + 
Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(2*b 
^2*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]])
 

Rubi [A] (warning: unable to verify)

Time = 1.77 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.09, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.130, Rules used = {3042, 4156, 3042, 4052, 27, 3042, 4132, 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.

\(\displaystyle \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot (c+d x)^{7/2} (a+b \tan (c+d x))^2}dx\)

\(\Big \downarrow \) 4156

\(\displaystyle \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {2 \int \frac {2 a b^3+\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) b^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {4 \int -\frac {b^2 \left (2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \int \frac {b^2 \left (2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \int \frac {2 a b-\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {2 \left (3 a^2+2 b^2\right )}{b d \sqrt {\cot (c+d x)}}-\frac {\frac {a^3 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 b^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}\)

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

Input:

Int[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2),x]
 

Output:

-(a^2/(b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x]))) + ((2*(3* 
a^2 + 2*b^2))/(b*d*Sqrt[Cot[c + d*x]]) - ((2*a^(5/2)*(3*a^2 + 7*b^2)*ArcTa 
n[(Sqrt[a]*Cot[c + d*x])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)*d) - (4*b^2*(-1/2* 
((a^2 - 2*a*b - b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + 
ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) + ((a^2 + 2*a*b - b^2)*(- 
1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + S 
qrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d 
))/b)/(2*b*(a^2 + b^2))
 

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 4052

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c 
+ d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 
/((m + 1)*(a^2 + b^2)*(b*c - a*d))   Int[(a + b*Tan[e + f*x])^(m + 1)*(c + 
d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - 
 a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / 
; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] 
 && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ 
erQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

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 4132

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*b^2 - a*(b*B - a*C))*(a + b*Tan[e + 
 f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + 
b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2))   Int[(a + b*Tan[e + 
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* 
(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d 
)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta 
n[e + f*x]^2, 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] && LtQ[m, -1] && 
!(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

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]
 

rule 4156

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Cot[e + f*x])^(m - n*p 
)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1053\) vs. \(2(265)=530\).

Time = 0.26 (sec) , antiderivative size = 1054, normalized size of antiderivative = 3.56

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1054\)
default	\(\text {Expression too large to display}\)	\(1054\)

Input:

int(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4/d*(8*tan(d*x+c)^(1/2)*(a*b)^(1/2)*a*b^4+16*tan(d*x+c)^(3/2)*(a*b)^(1/2 
)*a^2*b^3+8*tan(d*x+c)^(3/2)*(a*b)^(1/2)*a^4*b-12*arctan(b*tan(d*x+c)^(1/2 
)/(a*b)^(1/2))*a^5*b*tan(d*x+c)-28*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))* 
a^3*b^3*tan(d*x+c)+20*tan(d*x+c)^(1/2)*(a*b)^(1/2)*a^3*b^2+2*arctan(-1+2^( 
1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^2*b^3*tan(d*x+c)-4*arctan(-1+ 
2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b^4*tan(d*x+c)-2*ln(-(2^(1 
/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1) 
)*2^(1/2)*(a*b)^(1/2)*a*b^4*tan(d*x+c)-4*arctan(1+2^(1/2)*tan(d*x+c)^(1/2) 
)*2^(1/2)*(a*b)^(1/2)*a*b^4*tan(d*x+c)+ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^ 
(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*(a*b)^(1/2)*a^2* 
b^3*tan(d*x+c)+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^ 
2*b^3*tan(d*x+c)-12*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^6+ln(-(tan(d* 
x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))* 
2^(1/2)*(a*b)^(1/2)*a^3*b^2-ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2 
^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*(a*b)^(1/2)*a*b^4+2*arctan( 
1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^3*b^2-4*arctan(1+2^(1/2) 
*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^2*b^3-2*arctan(1+2^(1/2)*tan(d*x+ 
c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b^4+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))* 
2^(1/2)*(a*b)^(1/2)*a^3*b^2-4*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)* 
(a*b)^(1/2)*a^2*b^3-2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1557 vs. \(2 (265) = 530\).

Time = 0.50 (sec) , antiderivative size = 3140, normalized size of antiderivative = 10.61 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:

integrate(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(1/cot(d*x+c)**(7/2)/(a+b*tan(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, a^{5} + 7 \, a^{3} b^{2}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (2 \, a^{2} b + 2 \, b^{3} + \frac {3 \, a^{3} + 2 \, a b^{2}}{\tan \left (d x + c\right )}\right )}}{\frac {a^{2} b^{3} + b^{5}}{\sqrt {\tan \left (d x + c\right )}} + \frac {a^{3} b^{2} + a b^{4}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}}{4 \, d} \] Input:

integrate(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
 

Output:

1/4*(4*(3*a^5 + 7*a^3*b^2)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^4* 
b^2 + 2*a^2*b^4 + b^6)*sqrt(a*b)) - (2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan( 
1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 - 2*a*b - b 
^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt(2)*(a^2 + 
 2*a*b - b^2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt( 
2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 
1))/(a^4 + 2*a^2*b^2 + b^4) + 4*(2*a^2*b + 2*b^3 + (3*a^3 + 2*a*b^2)/tan(d 
*x + c))/((a^2*b^3 + b^5)/sqrt(tan(d*x + c)) + (a^3*b^2 + a*b^4)/tan(d*x + 
 c)^(3/2)))/d
 

Giac [F]

\[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate(1/((b*tan(d*x + c) + a)^2*cot(d*x + c)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \] Input:

int(1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2),x)
 

Output:

int(1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2), x)
 

Reduce [F]

\[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{4} \tan \left (d x +c \right )^{2} b^{2}+2 \cot \left (d x +c \right )^{4} \tan \left (d x +c \right ) a b +\cot \left (d x +c \right )^{4} a^{2}}d x \] Input:

int(1/cot(d*x+c)^(7/2)/(a+b*tan(d*x+c))^2,x)
 

Output:

int(sqrt(cot(c + d*x))/(cot(c + d*x)**4*tan(c + d*x)**2*b**2 + 2*cot(c + d 
*x)**4*tan(c + d*x)*a*b + cot(c + d*x)**4*a**2),x)