\(\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx\) [805]

Optimal result

Integrand size = 23, antiderivative size = 216 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d} \] Output:

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

Mathematica [C] (verified)

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

Time = 1.03 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {\frac {2}{7} a^2 \cot ^{\frac {7}{2}}(c+d x)+\frac {2}{3} \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )+\frac {1}{10} a b \left (-10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-40 \sqrt {\cot (c+d x)}+8 \cot ^{\frac {5}{2}}(c+d x)-5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{d} \] Input:

Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^2,x]
 

Output:

-(((2*a^2*Cot[c + d*x]^(7/2))/7 + (2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*(-1 + 
Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (a*b*(-10*Sqrt[2]*Ar 
cTan[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 + Sqr 
t[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/10)/d)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.14, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 4156, 3042, 4026, 3042, 4011, 3042, 4011, 3042, 4011, 3042, 4017, 25, 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.

\(\displaystyle \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (c+d x)^{9/2} (a+b \tan (c+d x))^2dx\)

\(\Big \downarrow \) 4156

\(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)^2dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

\(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) \left (-2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )dx-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-2 a b-\left (b^2-a^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\)

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (a^2+2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2\right )dx+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\)

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {4 a b \sqrt {\cot (c+d x)}}{d}\)

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^2,x]
 

Output:

(4*a*b*Sqrt[Cot[c + d*x]])/d + (2*(a^2 - b^2)*Cot[c + d*x]^(3/2))/(3*d) - 
(4*a*b*Cot[c + d*x]^(5/2))/(5*d) - (2*a^2*Cot[c + d*x]^(7/2))/(7*d) + (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 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/d
 

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 4026

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

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 [A] (verified)

Time = 0.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(-\frac {\frac {2 a^{2} \cot \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 a b \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 a^{2} \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-4 a b \sqrt {\cot \left (d x +c \right )}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\)	\(251\)
default	\(-\frac {\frac {2 a^{2} \cot \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 a b \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 a^{2} \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-4 a b \sqrt {\cot \left (d x +c \right )}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\)	\(251\)

Input:

int(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (185) = 370\).

Time = 0.13 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.36 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x, algorithm="fricas")
 

Output:

-1/210*(210*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d 
^2)*arctan(-(2*sqrt(1/2)*(a^2 - 2*a*b - b^2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2 
*b^2 - 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^4 + 4*a^3*b + 
2*a^2*b^2 - 4*a*b^3 + b^4)/d^2)*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 
+ b^4)/d^2))/(a^4 - 6*a^2*b^2 + b^4))*tan(d*x + c)^3 + 210*sqrt(1/2)*d*sqr 
t((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d^2)*arctan(-(2*sqrt(1/2)*(a 
^2 - 2*a*b - b^2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d^2)* 
sqrt(tan(d*x + c)) - d^2*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/ 
d^2)*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2))/(a^4 - 6*a^2*b 
^2 + b^4))*tan(d*x + c)^3 + 105*sqrt(1/2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b^ 
2 + 4*a*b^3 + b^4)/d^2)*log(2*sqrt(1/2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 
+ 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) - a^2 + 2*a*b + b^2 - (a^2 - 2*a* 
b - b^2)*tan(d*x + c))*tan(d*x + c)^3 - 105*sqrt(1/2)*d*sqrt((a^4 - 4*a^3* 
b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2)*log(-2*sqrt(1/2)*d*sqrt((a^4 - 4*a^3*b 
 + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) - a^2 + 2*a*b + b^2 
- (a^2 - 2*a*b - b^2)*tan(d*x + c))*tan(d*x + c)^3 - 4*(210*a*b*tan(d*x + 
c)^3 - 42*a*b*tan(d*x + c) + 35*(a^2 - b^2)*tan(d*x + c)^2 - 15*a^2)/sqrt( 
tan(d*x + c)))/(d*tan(d*x + c)^3)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**(9/2)*(a+b*tan(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.02 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {210 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {1680 \, a b}{\sqrt {\tan \left (d x + c\right )}} + \frac {336 \, a b}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {280 \, {\left (a^{2} - b^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {120 \, a^{2}}{\tan \left (d x + c\right )^{\frac {7}{2}}}}{420 \, d} \] Input:

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x, algorithm="maxima")
 

Output:

-1/420*(210*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sq 
rt(tan(d*x + c)))) + 210*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(-1/2*sqrt(2)*( 
sqrt(2) - 2/sqrt(tan(d*x + c)))) - 105*sqrt(2)*(a^2 - 2*a*b - b^2)*log(sqr 
t(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 105*sqrt(2)*(a^2 - 2*a*b - 
 b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 1680*a*b/sqr 
t(tan(d*x + c)) + 336*a*b/tan(d*x + c)^(5/2) - 280*(a^2 - b^2)/tan(d*x + c 
)^(3/2) + 120*a^2/tan(d*x + c)^(7/2))/d
 

Giac [F]

\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate((b*tan(d*x + c) + a)^2*cot(d*x + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \] Input:

int(cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^2,x)
 

Output:

int(cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^2, x)
 

Reduce [F]

\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{4} \tan \left (d x +c \right )^{2}d x \right ) b^{2}+2 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{4} \tan \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{4}d x \right ) a^{2} \] Input:

int(cot(d*x+c)^(9/2)*(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,x)*b**2 + 2*int(sqr 
t(cot(c + d*x))*cot(c + d*x)**4*tan(c + d*x),x)*a*b + int(sqrt(cot(c + d*x 
))*cot(c + d*x)**4,x)*a**2