\(\int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [35]

Optimal result

Integrand size = 34, antiderivative size = 223 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-8 a^4 (i A+B) x-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d} \] Output:

-8*a^4*(I*A+B)*x-8*a^4*(I*A+B)*cot(d*x+c)/d-4*a^4*(A-I*B)*cot(d*x+c)^2/d+1 
/60*a^4*(93*I*A+92*B)*cot(d*x+c)^3/d-8*a^4*(A-I*B)*ln(sin(d*x+c))/d-1/6*a* 
A*cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3/d-1/10*(3*I*A+2*B)*cot(d*x+c)^5*(a^2+I 
*a^2*tan(d*x+c))^2/d+1/20*(13*A-12*I*B)*cot(d*x+c)^4*(a^4+I*a^4*tan(d*x+c) 
)/d
 

Mathematica [A] (verified)

Time = 2.44 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 \left ((13 A-12 i B) (i+\cot (c+d x))^4-4 i (2 A-3 i B) \cot (c+d x) (i+\cot (c+d x))^4-10 A \cot ^2(c+d x) (i+\cot (c+d x))^4+20 (i A+B) \left (-21 \cot (c+d x)+6 i \cot ^2(c+d x)+\cot ^3(c+d x)+24 i (\log (\tan (c+d x))-\log (i+\tan (c+d x)))\right )\right )}{60 d} \] Input:

Integrate[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]
 

Output:

(a^4*((13*A - (12*I)*B)*(I + Cot[c + d*x])^4 - (4*I)*(2*A - (3*I)*B)*Cot[c 
 + d*x]*(I + Cot[c + d*x])^4 - 10*A*Cot[c + d*x]^2*(I + Cot[c + d*x])^4 + 
20*(I*A + B)*(-21*Cot[c + d*x] + (6*I)*Cot[c + d*x]^2 + Cot[c + d*x]^3 + ( 
24*I)*(Log[Tan[c + d*x]] - Log[I + Tan[c + d*x]]))))/(60*d)
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.10, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.618, Rules used = {3042, 4076, 27, 3042, 4076, 27, 3042, 4076, 3042, 4074, 27, 3042, 4012, 3042, 4012, 25, 3042, 4014, 3042, 25, 3956}

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.

\(\displaystyle \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (c+d x))^4 (A+B \tan (c+d x))}{\tan (c+d x)^7}dx\)

\(\Big \downarrow \) 4076

\(\displaystyle \frac {1}{6} \int 3 \cot ^6(c+d x) (i \tan (c+d x) a+a)^3 (a (3 i A+2 B)-a (A-2 i B) \tan (c+d x))dx-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \cot ^6(c+d x) (i \tan (c+d x) a+a)^3 (a (3 i A+2 B)-a (A-2 i B) \tan (c+d x))dx-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {(i \tan (c+d x) a+a)^3 (a (3 i A+2 B)-a (A-2 i B) \tan (c+d x))}{\tan (c+d x)^6}dx-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4076

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int -2 \cot ^5(c+d x) (i \tan (c+d x) a+a)^2 \left ((13 A-12 i B) a^2+(7 i A+8 B) \tan (c+d x) a^2\right )dx-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \int \cot ^5(c+d x) (i \tan (c+d x) a+a)^2 \left ((13 A-12 i B) a^2+(7 i A+8 B) \tan (c+d x) a^2\right )dx-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \int \frac {(i \tan (c+d x) a+a)^2 \left ((13 A-12 i B) a^2+(7 i A+8 B) \tan (c+d x) a^2\right )}{\tan (c+d x)^5}dx-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4076

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \int \cot ^4(c+d x) (i \tan (c+d x) a+a) \left (a^3 (93 i A+92 B)-a^3 (67 A-68 i B) \tan (c+d x)\right )dx-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \int \frac {(i \tan (c+d x) a+a) \left (a^3 (93 i A+92 B)-a^3 (67 A-68 i B) \tan (c+d x)\right )}{\tan (c+d x)^4}dx-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4074

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (\int -160 \cot ^3(c+d x) \left ((A-i B) a^4+(i A+B) \tan (c+d x) a^4\right )dx-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \int \cot ^3(c+d x) \left ((A-i B) a^4+(i A+B) \tan (c+d x) a^4\right )dx-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \int \frac {(A-i B) a^4+(i A+B) \tan (c+d x) a^4}{\tan (c+d x)^3}dx-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (\int \cot ^2(c+d x) \left (a^4 (i A+B)-a^4 (A-i B) \tan (c+d x)\right )dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (\int \frac {a^4 (i A+B)-a^4 (A-i B) \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (\int -\cot (c+d x) \left ((A-i B) a^4+(i A+B) \tan (c+d x) a^4\right )dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (-\int \cot (c+d x) \left ((A-i B) a^4+(i A+B) \tan (c+d x) a^4\right )dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (-\int \frac {(A-i B) a^4+(i A+B) \tan (c+d x) a^4}{\tan (c+d x)}dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4014

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (-a^4 (A-i B) \int \cot (c+d x)dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}-a^4 x (B+i A)\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (-a^4 (A-i B) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}-a^4 x (B+i A)\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-160 \left (a^4 (A-i B) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}-a^4 x (B+i A)\right )-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\frac {1}{4} \left (-\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{3 d}-160 \left (-\frac {a^4 (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a^4 (B+i A) \cot (c+d x)}{d}-\frac {a^4 (A-i B) \log (-\sin (c+d x))}{d}-a^4 x (B+i A)\right )\right )-\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{4 d}\right )-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}\)

Input:

Int[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]
 

Output:

-1/6*(a*A*Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3)/d + (-1/5*(((3*I)*A + 2 
*B)*Cot[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^2)/d - (2*((-1/3*(a^4*((93*I 
)*A + 92*B)*Cot[c + d*x]^3)/d - 160*(-(a^4*(I*A + B)*x) - (a^4*(I*A + B)*C 
ot[c + d*x])/d - (a^4*(A - I*B)*Cot[c + d*x]^2)/(2*d) - (a^4*(A - I*B)*Log 
[-Sin[c + d*x]])/d))/4 - ((13*A - (12*I)*B)*Cot[c + d*x]^4*(a^4 + I*a^4*Ta 
n[c + d*x]))/(4*d)))/5)/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 3042

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

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4012

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

rule 4014

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a 
*d)/(a^2 + b^2)   Int[(b - a*Tan[e + f*x])/(a + b*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] && N 
eQ[a*c + b*d, 0]
 

rule 4074

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

rule 4076

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

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(\frac {8 \left (\frac {\left (-i B +A \right ) \ln \left (\sec \left (d x +c \right )^{2}\right )}{2}+\left (i B -A \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \cot \left (d x +c \right )^{6}}{48}-\frac {\cot \left (d x +c \right )^{5} \left (i A +\frac {B}{4}\right )}{10}+\frac {\left (-i B +\frac {7 A}{4}\right ) \cot \left (d x +c \right )^{4}}{8}+\frac {\left (i A +\frac {7 B}{8}\right ) \cot \left (d x +c \right )^{3}}{3}+\frac {\cot \left (d x +c \right )^{2} \left (i B -A \right )}{2}-\cot \left (d x +c \right ) \left (i A +B \right )-\left (i A +B \right ) x d \right ) a^{4}}{d}\)	\(150\)
derivativedivides	\(\frac {a^{4} \left (\frac {\left (-8 i B +8 A \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-8 i A -8 B \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{6 \tan \left (d x +c \right )^{6}}+\left (8 i B -8 A \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {-8 i B +8 A}{2 \tan \left (d x +c \right )^{2}}-\frac {8 i A +8 B}{\tan \left (d x +c \right )}-\frac {-8 i A -7 B}{3 \tan \left (d x +c \right )^{3}}-\frac {4 i A +B}{5 \tan \left (d x +c \right )^{5}}-\frac {4 i B -7 A}{4 \tan \left (d x +c \right )^{4}}\right )}{d}\)	\(161\)
default	\(\frac {a^{4} \left (\frac {\left (-8 i B +8 A \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-8 i A -8 B \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{6 \tan \left (d x +c \right )^{6}}+\left (8 i B -8 A \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {-8 i B +8 A}{2 \tan \left (d x +c \right )^{2}}-\frac {8 i A +8 B}{\tan \left (d x +c \right )}-\frac {-8 i A -7 B}{3 \tan \left (d x +c \right )^{3}}-\frac {4 i A +B}{5 \tan \left (d x +c \right )^{5}}-\frac {4 i B -7 A}{4 \tan \left (d x +c \right )^{4}}\right )}{d}\)	\(161\)
risch	\(\frac {16 a^{4} B c}{d}+\frac {16 i a^{4} A c}{d}-\frac {4 i a^{4} \left (270 i A \,{\mathrm e}^{10 i \left (d x +c \right )}+210 B \,{\mathrm e}^{10 i \left (d x +c \right )}-855 i A \,{\mathrm e}^{8 i \left (d x +c \right )}-765 B \,{\mathrm e}^{8 i \left (d x +c \right )}+1350 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+1210 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1125 i A \,{\mathrm e}^{4 i \left (d x +c \right )}-1020 B \,{\mathrm e}^{4 i \left (d x +c \right )}+486 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+444 B \,{\mathrm e}^{2 i \left (d x +c \right )}-86 i A -79 B \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {8 A \,a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(221\)
norman	\(\frac {\left (-8 i A \,a^{4}-8 B \,a^{4}\right ) x \tan \left (d x +c \right )^{6}-\frac {A \,a^{4}}{6 d}+\frac {\left (-4 i B \,a^{4}+7 A \,a^{4}\right ) \tan \left (d x +c \right )^{2}}{4 d}-\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )}{5 d}-\frac {4 \left (-i B \,a^{4}+A \,a^{4}\right ) \tan \left (d x +c \right )^{4}}{d}-\frac {8 \left (i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )^{5}}{d}+\frac {\left (8 i A \,a^{4}+7 B \,a^{4}\right ) \tan \left (d x +c \right )^{3}}{3 d}}{\tan \left (d x +c \right )^{6}}-\frac {8 \left (-i B \,a^{4}+A \,a^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {4 \left (-i B \,a^{4}+A \,a^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\)	\(229\)

Input:

int(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVER 
BOSE)
 

Output:

8*(1/2*(A-I*B)*ln(sec(d*x+c)^2)+(-A+I*B)*ln(tan(d*x+c))-1/48*A*cot(d*x+c)^ 
6-1/10*cot(d*x+c)^5*(I*A+1/4*B)+1/8*(-I*B+7/4*A)*cot(d*x+c)^4+1/3*(I*A+7/8 
*B)*cot(d*x+c)^3+1/2*cot(d*x+c)^2*(-A+I*B)-cot(d*x+c)*(I*A+B)-(I*A+B)*x*d) 
*a^4/d
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.49 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {4 \, {\left (30 \, {\left (9 \, A - 7 i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 \, {\left (19 \, A - 17 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, {\left (135 \, A - 121 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 15 \, {\left (75 \, A - 68 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, {\left (81 \, A - 74 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (86 \, A - 79 i \, B\right )} a^{4} - 30 \, {\left ({\left (A - i \, B\right )} a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, {\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, {\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, {\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm= 
"fricas")
 

Output:

4/15*(30*(9*A - 7*I*B)*a^4*e^(10*I*d*x + 10*I*c) - 45*(19*A - 17*I*B)*a^4* 
e^(8*I*d*x + 8*I*c) + 10*(135*A - 121*I*B)*a^4*e^(6*I*d*x + 6*I*c) - 15*(7 
5*A - 68*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 6*(81*A - 74*I*B)*a^4*e^(2*I*d*x + 
 2*I*c) - (86*A - 79*I*B)*a^4 - 30*((A - I*B)*a^4*e^(12*I*d*x + 12*I*c) - 
6*(A - I*B)*a^4*e^(10*I*d*x + 10*I*c) + 15*(A - I*B)*a^4*e^(8*I*d*x + 8*I* 
c) - 20*(A - I*B)*a^4*e^(6*I*d*x + 6*I*c) + 15*(A - I*B)*a^4*e^(4*I*d*x + 
4*I*c) - 6*(A - I*B)*a^4*e^(2*I*d*x + 2*I*c) + (A - I*B)*a^4)*log(e^(2*I*d 
*x + 2*I*c) - 1))/(d*e^(12*I*d*x + 12*I*c) - 6*d*e^(10*I*d*x + 10*I*c) + 1 
5*d*e^(8*I*d*x + 8*I*c) - 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x + 4*I 
*c) - 6*d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [A] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.56 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=- \frac {8 a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 344 A a^{4} + 316 i B a^{4} + \left (1944 A a^{4} e^{2 i c} - 1776 i B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 4500 A a^{4} e^{4 i c} + 4080 i B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (5400 A a^{4} e^{6 i c} - 4840 i B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 3420 A a^{4} e^{8 i c} + 3060 i B a^{4} e^{8 i c}\right ) e^{8 i d x} + \left (1080 A a^{4} e^{10 i c} - 840 i B a^{4} e^{10 i c}\right ) e^{10 i d x}}{15 d e^{12 i c} e^{12 i d x} - 90 d e^{10 i c} e^{10 i d x} + 225 d e^{8 i c} e^{8 i d x} - 300 d e^{6 i c} e^{6 i d x} + 225 d e^{4 i c} e^{4 i d x} - 90 d e^{2 i c} e^{2 i d x} + 15 d} \] Input:

integrate(cot(d*x+c)**7*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)
 

Output:

-8*a**4*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-344*A*a**4 + 316*I 
*B*a**4 + (1944*A*a**4*exp(2*I*c) - 1776*I*B*a**4*exp(2*I*c))*exp(2*I*d*x) 
 + (-4500*A*a**4*exp(4*I*c) + 4080*I*B*a**4*exp(4*I*c))*exp(4*I*d*x) + (54 
00*A*a**4*exp(6*I*c) - 4840*I*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (-3420*A*a 
**4*exp(8*I*c) + 3060*I*B*a**4*exp(8*I*c))*exp(8*I*d*x) + (1080*A*a**4*exp 
(10*I*c) - 840*I*B*a**4*exp(10*I*c))*exp(10*I*d*x))/(15*d*exp(12*I*c)*exp( 
12*I*d*x) - 90*d*exp(10*I*c)*exp(10*I*d*x) + 225*d*exp(8*I*c)*exp(8*I*d*x) 
 - 300*d*exp(6*I*c)*exp(6*I*d*x) + 225*d*exp(4*I*c)*exp(4*I*d*x) - 90*d*ex 
p(2*I*c)*exp(2*I*d*x) + 15*d)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.77 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {480 \, {\left (d x + c\right )} {\left (i \, A + B\right )} a^{4} - 240 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac {480 \, {\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )^{5} - 240 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} + 20 \, {\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} + 15 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 12 \, {\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) - 10 \, A a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \] Input:

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm= 
"maxima")
 

Output:

-1/60*(480*(d*x + c)*(I*A + B)*a^4 - 240*(A - I*B)*a^4*log(tan(d*x + c)^2 
+ 1) + 480*(A - I*B)*a^4*log(tan(d*x + c)) - (480*(-I*A - B)*a^4*tan(d*x + 
 c)^5 - 240*(A - I*B)*a^4*tan(d*x + c)^4 + 20*(8*I*A + 7*B)*a^4*tan(d*x + 
c)^3 + 15*(7*A - 4*I*B)*a^4*tan(d*x + c)^2 + 12*(-4*I*A - B)*a^4*tan(d*x + 
 c) - 10*A*a^4)/tan(d*x + c)^6)/d
 

Giac [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.82 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {8 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{d} - \frac {8 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {480 i \, {\left (A a^{4} - i \, B a^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} - 240 i \, {\left (i \, A a^{4} + B a^{4}\right )} \tan \left (d x + c\right )^{4} - 20 i \, {\left (8 \, A a^{4} - 7 i \, B a^{4}\right )} \tan \left (d x + c\right )^{3} - 15 i \, {\left (-7 i \, A a^{4} - 4 \, B a^{4}\right )} \tan \left (d x + c\right )^{2} + 12 i \, {\left (4 \, A a^{4} - i \, B a^{4}\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{6}} \] Input:

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm= 
"giac")
 

Output:

8*(A*a^4 - I*B*a^4)*log(tan(d*x + c) + I)/d - 8*(A*a^4 - I*B*a^4)*log(abs( 
tan(d*x + c)))/d - 1/60*(480*I*(A*a^4 - I*B*a^4)*tan(d*x + c)^5 + 10*A*a^4 
 - 240*I*(I*A*a^4 + B*a^4)*tan(d*x + c)^4 - 20*I*(8*A*a^4 - 7*I*B*a^4)*tan 
(d*x + c)^3 - 15*I*(-7*I*A*a^4 - 4*B*a^4)*tan(d*x + c)^2 + 12*I*(4*A*a^4 - 
 I*B*a^4)*tan(d*x + c))/(d*tan(d*x + c)^6)
 

Mupad [B] (verification not implemented)

Time = 4.82 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.73 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (4\,A\,a^4-B\,a^4\,4{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A\,a^4}{4}-B\,a^4\,1{}\mathrm {i}\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (8\,B\,a^4+A\,a^4\,8{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {7\,B\,a^4}{3}+\frac {A\,a^4\,8{}\mathrm {i}}{3}\right )+\frac {A\,a^4}{6}+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{5}+\frac {A\,a^4\,4{}\mathrm {i}}{5}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^6}-\frac {16\,a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{d} \] Input:

int(cot(c + d*x)^7*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)
 

Output:

- (tan(c + d*x)^4*(4*A*a^4 - B*a^4*4i) - tan(c + d*x)^2*((7*A*a^4)/4 - B*a 
^4*1i) + tan(c + d*x)^5*(A*a^4*8i + 8*B*a^4) - tan(c + d*x)^3*((A*a^4*8i)/ 
3 + (7*B*a^4)/3) + (A*a^4)/6 + tan(c + d*x)*((A*a^4*4i)/5 + (B*a^4)/5))/(d 
*tan(c + d*x)^6) - (16*a^4*atan(2*tan(c + d*x) + 1i)*(A*1i + B))/d
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.39 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^{4} \left (-1376 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a i -1264 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b +512 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a i +328 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -96 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a i -24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6} a -960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6} b i -960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a +960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} b i -960 \sin \left (d x +c \right )^{6} a d i x +385 \sin \left (d x +c \right )^{6} a -960 \sin \left (d x +c \right )^{6} b d x -315 \sin \left (d x +c \right )^{6} b i -960 \sin \left (d x +c \right )^{4} a +720 \sin \left (d x +c \right )^{4} b i +270 \sin \left (d x +c \right )^{2} a -120 \sin \left (d x +c \right )^{2} b i -20 a \right )}{120 \sin \left (d x +c \right )^{6} d} \] Input:

int(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)
 

Output:

(a**4*( - 1376*cos(c + d*x)*sin(c + d*x)**5*a*i - 1264*cos(c + d*x)*sin(c 
+ d*x)**5*b + 512*cos(c + d*x)*sin(c + d*x)**3*a*i + 328*cos(c + d*x)*sin( 
c + d*x)**3*b - 96*cos(c + d*x)*sin(c + d*x)*a*i - 24*cos(c + d*x)*sin(c + 
 d*x)*b + 960*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6*a - 960*log(tan 
((c + d*x)/2)**2 + 1)*sin(c + d*x)**6*b*i - 960*log(tan((c + d*x)/2))*sin( 
c + d*x)**6*a + 960*log(tan((c + d*x)/2))*sin(c + d*x)**6*b*i - 960*sin(c 
+ d*x)**6*a*d*i*x + 385*sin(c + d*x)**6*a - 960*sin(c + d*x)**6*b*d*x - 31 
5*sin(c + d*x)**6*b*i - 960*sin(c + d*x)**4*a + 720*sin(c + d*x)**4*b*i + 
270*sin(c + d*x)**2*a - 120*sin(c + d*x)**2*b*i - 20*a))/(120*sin(c + d*x) 
**6*d)